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Short Tail Keywords textstyle sigma normal |
long Tail Keywords (2 words) normal distribution standard normal xtextstyle x sigma 2 textstyle mu |
long Tail Keywords (3 words) standard normal distribution cumulative distribution function 2textstyle sigma 2 xtextstyle phi x mean textstyle mu taylor series expansion variable xtextstyle x |
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wikipedia free encyclopedia displaystyle mathcal nmu sigma mathbb xin frac sqrt efrac xmu phi leftfrac rightfrac left operatorname erf rightright log esigma expmu tsigma expimu imu beginpmatrix endpmatrix over leftleftfrac right leftqpleftfrac fxfrac textstyle varphi zfrac zepi fxmid xsigma zmu zxmu xsim tau fxsqrt etau theta xfrac int infty xet xxet dtfrac xsqrt fxphi pxx xdxxphi xvarphi cdot leftxfrac cdots xapprox sum nfrac eax xsum nxx beginalignedphi dtphi leftx rightngeq endaligned textdesired xnfrac xnx textdesiredphi exn nsigma fmu nphi noperatorname nsqrt ldots pfmu textor text papprox psqrt quad pin pmu zpsigma pphi xxvarphi nxvarphi leftxmu prightbegincases textif ptext oddsigma evenendcases beginalignedoperatorname prightsigma begincasessqrt odd evenendcasessigma pcdot gamma rightsqrt neq leftxprightsigma isqrt puleftfrac rightoperatorname rightendaligned leftxmid axbrightmu fbfafbfa binfty hat ftint fxeitxdxeimu tefrac eitx xthat etx mtoperatorname leftetxrighthat fitemu tesigma tkk exk gtln mtmu ttfrac afxsigma fxxmu fmathbb mbox such fxinfty xneq delta fxdelta fxbegincases xgeq endcases hxint fxln fxdx fxlog lint fxdxlambda fxdxrightlambda leftmu fxxdxrightlambda leftsigma dxright fxleftln lambda xlambda rightdx fxexp hxtfrac xtexp sim dmathrm parallel exp beginpmatrixfrac mid xnsim nleftfrac bar nabla dots emax ixileq zsqrt nsum nxiright bnp chi tnu cos rho vert xivert axb amu exsim lnnmu pmathcal leftxrightsim nfmu pxfrac abmu alpha xksim nmksigma kin xtextdxx nmalpha malpha fzzpi cauchy tfrac overline ssqrt xnmu xnoverline rightsim ffrac rightnlefty rightmsim fnm qsum xjc textn akxk bkxk akbksigma fxxnmu xleq xleft rightsigma lmu nln fximid nximu xequiv nxiqquad nxioverline lhat nrightarrow nhat xrightarrow qquad ncdot simeq xioverline lefthat zalpha lefts zkxkhat axy bxz ableftxfrac aybzabright ababyz aybzab ababfrac afrac abab ktimes beginalignedmathbf mathbf ijaijxixj xixjxjxi aijaji nxibar nbar nxi beginalignedpmathbf prod ximu rightleftfrac rightn leftsum rightrightendaligned beginalignedpmu propto pmathbf rightrightsqrt rightpropto lefttau righttau rightrightpropto leftntau rightrightexp ntau dfrac xtau beginalignedtau ptmu ptbar nxiendaligned beginalignedsigma psigma ssum beginalignedpsigma beginalignednu nnu nendaligned beginalignedalpha beta ichi ignu beginalignedbar rightpsigma leftnu leftsnbar rightrightsigma sfrac nleftmu nright rightquad times rightrightmathcal nrightcdot igsigma partial tfxtfrac fxt ucos vqquad ysqrt usin xusqrt ssqquad yvsqrt xleftb rightvarepsilon xqquad xleftxfrac leftapprox beginaligned leftxrightleftfrac rightefrac leftxright leftxrightright zphi ppright rightqquad pgeq pplog beginalignedlzint zinfty uzvarphi uduint udu ptlzapprox begincases leftdfrac prightzp pprightpgeq pttextor equivalentlylzapprox prightrightp pppgeq endcasesendaligned ell mathit hsurd emathrm dffrac exm icon xalpha expbeta xpsi psi leftbeginmatrixleftalpha endmatrixzright edit wikidata wikimedia foundation powered mediawiki
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SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
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PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 87% | A title should reflect the contents of a site. This site has a 67 % match | |
Title Length | 80% | Limit your title to anywhere between 40 and 70 characters. Your title was 32 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
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Number of internal links | 30% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 961 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 4 level 1 folders and 8 folders above or in the first level of navigation. | |
Headings | 32% | Headers should reflect the contents of a site. This site has a 14 % match | |
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Page errors | 100% | Pages with no errors display significantly faster on most browsers. We detected 0 errors and warnings | |
WordCount | 20% | An ideal page contains between 400 and 600 words.This page contains 20104 words | |
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Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 1423 inline style declarations ( <a style="color:green">) with a size of 47520 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
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Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
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Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
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http://en.wikipedia.org/static/images/icons/wikipedia.png height: 50 width: 50 description: no alt description found |
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http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en.svg height: height attribute not set width: width attribute not set description: wikipedia |
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http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en.svg height: 13 width: 117 description: the free encyclopedia |
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https://upload.wikimedia.org/wikipedia/commons/thumb/7/74/normal_distribution_pdf.svg/400px-normal_distribution_pdf.svg.png height: 256 width: 400 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/normal_distribution_cdf.svg/400px-normal_distribution_cdf.svg.png height: 256 width: 400 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {n}}(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a48f0e84328dc53dec2ad301bb321c00dcf422 height: height attribute not set width: width attribute not set description: {\displaystyle \mu \in \mathbb {r} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5b81059aaa2db87802908711c7527c0e1d65fc06 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{2}\in \mathbb {r} _{>0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab height: height attribute not set width: width attribute not set description: {\displaystyle x\in \mathbb {r} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce7e315b02666699e0cd8ea5fb1a3e0c287cd9d height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fed43e25966344745178c406f04b15d0fa3783 height: height attribute not set width: width attribute not set description: {\displaystyle \phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/510cf4025ba141645f21c7f06866867648d5bc21 height: height attribute not set width: width attribute not set description: {\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161 height: height attribute not set width: width attribute not set description: {\displaystyle \mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3712c4b0b190f29a0c70285b08a63593ef3e1885 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma {\sqrt {2/\pi }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950 height: height attribute not set width: width attribute not set description: {\displaystyle 0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c62185fa421e99aeb74288e52e9b8d703cbca074 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{2}}\log(2\pi e\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0b73a531ccd6d15d03c0b80f4707641f57bfd521 height: height attribute not set width: width attribute not set description: {\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8876b3855a069ca3b9464f56ffd8ab9c4b191c50 height: height attribute not set width: width attribute not set description: {\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ffca6c1c08385fa2c44455f67e41efb357d3ad8c height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {i}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2047e1545c0f223d425a34cb783fb2928a87dc2e height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {i}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1668ab76d345b5cd802c87a9cd3c4f8e735f30 height: height attribute not set width: width attribute not set description: {\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+\ln {\sigma _{1}^{2} \over \sigma _{0}^{2}}\right\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1f49431e199e2ecbff74ff719606369663cf50f4 height: height attribute not set width: width attribute not set description: {\displaystyle \mu -\sigma {\frac {{\frac {1}{\sqrt {2\pi }}}e^{\frac {-\left(q_{p}\left({\frac {x-\mu }{\sigma }}\right)\right)^{2}}{2}}}{1-p}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/3/3a/standard_deviation_diagram_micro.svg/250px-standard_deviation_diagram_micro.svg.png height: 125 width: 250 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac6c71a4a3df62eeaf7e052e27ce356793102f5 height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/259577540a13444806174d5a1ae7662974f58085 height: height attribute not set width: width attribute not set description: {\textstyle \mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a86f1d00f664920ef46109bcddc0778f4976b490 height: height attribute not set width: width attribute not set description: {\textstyle \sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca336f17488923d11206e131f7eb2a569e8dde2 height: height attribute not set width: width attribute not set description: {\textstyle \sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/501d35ed7f32d2adb071cbde62acd5fe6218d118 height: height attribute not set width: width attribute not set description: {\textstyle \mu =0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2598ad7a8b77472fe5859d97603776dce7d5ba height: height attribute not set width: width attribute not set description: {\textstyle \sigma ^{2}=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aef3a1ca677d83f35a5b3a72a38d9788279a28d4 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi (z)={\frac {e^{\frac {-z^{2}}{2}}}{\sqrt {2\pi }}}\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7a33e37010e3acdeeb80fdb95df9bfe411fd79e6 height: height attribute not set width: width attribute not set description: {\textstyle z} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1358bdf1286826fffd8e70843233399b5d69f8ee height: height attribute not set width: width attribute not set description: {\textstyle \varphi (z)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eccd2c28dc343be5631094e573191b2c17edd21d height: height attribute not set width: width attribute not set description: {\textstyle {\frac {1}{\sqrt {2\pi }}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0829ff59a6fdc19b44396956e8767fac4ba87ba3 height: height attribute not set width: width attribute not set description: {\textstyle z=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/696b845d6e60244a38cdcbe4c380cf780fa896cb height: height attribute not set width: width attribute not set description: {\textstyle z=+1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b192f77620e50d9d1f9d417be0fbcd2813ab4b height: height attribute not set width: width attribute not set description: {\textstyle z=-1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5d08fa7405690b759b8c863dfd86313b7bfe1471 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/67d6310ec49a504c34d6800009a9467b8f5302e2 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi (z)=e^{-\pi z^{2}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7925395446d59209f7c582b61decbda1ff2c5fea height: height attribute not set width: width attribute not set description: {\textstyle \sigma ^{2}=1/(2\pi ).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fd10de785e04c8023943b58c3794881d6037d959 height: height attribute not set width: width attribute not set description: {\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d20071b089d5bd816234098110b4c2bd18ed32d9 height: height attribute not set width: width attribute not set description: {\textstyle 1/\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d442d6fe240625e70b574360ee971066244df646 height: height attribute not set width: width attribute not set description: {\textstyle z} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1acabbc54b0e79a7c76dce7de93aaaee9374b206 height: height attribute not set width: width attribute not set description: {\textstyle x=\sigma z+\mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391 height: height attribute not set width: width attribute not set description: {\textstyle x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2807a7efd142e91103b494889fff5f3c4f3b56 height: height attribute not set width: width attribute not set description: {\textstyle z=(x-\mu )/\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10 height: height attribute not set width: width attribute not set description: {\textstyle \phi } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/99015519246670af1cb5592e439ad64a27fb4830 height: height attribute not set width: width attribute not set description: {\textstyle \varphi } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6c34ca495ee2609c49ba6b010c03c31e1968ae87 height: height attribute not set width: width attribute not set description: {\textstyle n(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fa40efb531f5b9513c921fd804868e727dfc71c0 height: height attribute not set width: width attribute not set description: {\textstyle {\mathcal {n}}(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0aeea1216143061c89f6a1944928a0aeee1b9cb1 height: height attribute not set width: width attribute not set description: {\displaystyle x\sim {\mathcal {n}}(\mu ,\sigma ^{2}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d590a3e8735feb2d65c6fa0c4bc71ff946cd8bec height: height attribute not set width: width attribute not set description: {\textstyle \tau } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b4bd5cc088e8ac985203e3e55ee1d75cf6ce32d8 height: height attribute not set width: width attribute not set description: {\textstyle 1/\sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e18260c517de859f1451477bc0c91d2d46f092a1 height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb4526c1105a954e898b1a04d9614d40eb53a98 height: height attribute not set width: width attribute not set description: {\textstyle \tau '=1/\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e16b2715bf35fcc08b305f756ca921d8557ddc4f height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\frac {\tau '}{\sqrt {2\pi }}}e^{-(\tau ')^{2}(x-\mu )^{2}/2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/008c5a8c14b3669bb14c1af1053414284aa96206 height: height attribute not set width: width attribute not set description: {\textstyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/25441db9629d5369d06da54d4eed34af4bc9a578 height: height attribute not set width: width attribute not set description: {\textstyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9d3972646af30b9fed0b85290e7e1956a8799096 height: height attribute not set width: width attribute not set description: {\textstyle \phi } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe155a8dcfd41749d5e0e0e6a9ee8bc2614b32b5 height: height attribute not set width: width attribute not set description: {\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/90f3dc9d94749c443d5adce54b6a51200bdd52f8 height: height attribute not set width: width attribute not set description: {\textstyle \operatorname {erf} (x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4939f2eb7c55ff4e7d13331fc0599a5f2452b479 height: height attribute not set width: width attribute not set description: {\textstyle [-x,x]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/42ad512624379200c07b7148812f0841710e5650 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4dec6de19ebeaa734086622f167d87ac41e7b50d height: height attribute not set width: width attribute not set description: {\displaystyle \phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283 height: height attribute not set width: width attribute not set description: {\textstyle f} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f48cf2d3b65754e434e6a6c5a90908dd83d44f12 height: height attribute not set width: width attribute not set description: {\displaystyle f(x)=\phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/55ee5d81b5e0df283ceaf4c4fb84f2735fa703e4 height: height attribute not set width: width attribute not set description: {\textstyle q(x)=1-\phi (x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781 height: height attribute not set width: width attribute not set description: {\textstyle x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bfaae98effc2a25e28a050e520e5f78d3b9a17f8 height: height attribute not set width: width attribute not set description: {\textstyle p(x>x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/131bb2da649dd6c113517c5ae8c26370654ee8fa height: height attribute not set width: width attribute not set description: {\textstyle q} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/171e231717561bbe784ad6b435005303fdc317ec height: height attribute not set width: width attribute not set description: {\textstyle \phi (-x)=1-\phi (x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec7f747ce873d091260c617c82359d7c407fee6 height: height attribute not set width: width attribute not set description: {\displaystyle \int \phi (x)\,dx=x\phi (x)+\varphi (x)+c.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ad8275ec6180f68c2dae03c9e70716b75ff4f3aa height: height attribute not set width: width attribute not set description: {\displaystyle \phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/89dd9aa9898e3a45a024e12e27086d636a1bf5cd height: height attribute not set width: width attribute not set description: {\textstyle !!} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/be5ebedd65f675d03340c2b762d86b7a28fe4092 height: height attribute not set width: width attribute not set description: {\displaystyle \phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/21938180273579358f8b56792c925140a02edc7c height: height attribute not set width: width attribute not set description: {\textstyle e^{ax^{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/395212d2e4fbaa127006c9492b68cbb86e06301e height: height attribute not set width: width attribute not set description: {\textstyle \phi (x_{0})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/899b25343db480efb1ae8a44a5331e78497b83cd height: height attribute not set width: width attribute not set description: {\displaystyle \phi (x)=\sum _{n=0}^{\infty }{\frac {\phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\,,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fed2112b27830b104cb3df5ea89238adbca49314 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\phi ^{(0)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt\\\phi ^{(1)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}\\\phi ^{(n)}(x_{0})&=-\left(x_{0}\phi ^{(n-1)}(x_{0})+(n-2)\phi ^{(n-2)}(x_{0})\right),&n\geq 2\,.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fb97785cbcee6a45a8885563bcc90a913b803a87 height: height attribute not set width: width attribute not set description: {\textstyle \phi (x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/29af9ce0dc59266a05d1fd7820ab6f2f0ecd9c3e height: height attribute not set width: width attribute not set description: {\textstyle x_{0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9dff36c86b3ebfd445e97f6fc29ac32d4c57e0 height: height attribute not set width: width attribute not set description: {\textstyle \phi (x_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd28344f1edbe2c8ad3af7ff95f52bcc88fab27 height: height attribute not set width: width attribute not set description: {\textstyle \phi ({\text{desired}})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/80cc6b7973121e0f8c235d296fae5639bf54c7e6 height: height attribute not set width: width attribute not set description: {\displaystyle x_{n+1}=x_{n}-{\frac {\phi (x_{n},x_{0},\phi (x_{0}))-\phi ({\text{desired}})}{\phi '(x_{n})}}\,,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9a93e44f156df5395f7acd49141a4fda94825d height: height attribute not set width: width attribute not set description: {\textstyle \phi (x,x_{0},\phi (x_{0}))} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/49a5822737668c8887901648dcd7461ee18221cc height: height attribute not set width: width attribute not set description: {\displaystyle \phi '(x_{n})={\frac {1}{\sqrt {2\pi }}}e^{-x_{n}^{2}/2}\,.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/standard_deviation_diagram.svg/350px-standard_deviation_diagram.svg.png height: 175 width: 350 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/16ab399c484083545cd1266951f08743135539f3 height: height attribute not set width: width attribute not set description: {\textstyle \mu -n\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d5a81dd97c468e3806e43e15761506633efb95 height: height attribute not set width: width attribute not set description: {\textstyle \mu +n\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/effeceb477bf37b05d0035347946350b1f0155ce height: height attribute not set width: width attribute not set description: {\displaystyle f(\mu +n\sigma )-f(\mu -n\sigma )=\phi (n)-\phi (-n)=\operatorname {erf} \left({\frac {n}{\sqrt {2}}}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/79a9883d3cba69db9041e545b5bd90383dc72a76 height: height attribute not set width: width attribute not set description: {\textstyle n=1,2,\ldots ,6} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6 height: height attribute not set width: width attribute not set description: {\textstyle n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a57a52e2d443b77812c701fe2da3926fb7d371c1 height: height attribute not set width: width attribute not set description: {\textstyle p=f(\mu +n\sigma )-f(\mu -n\sigma )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ec5d41b23d96cef795a32f25060ef6ba352876bd height: height attribute not set width: width attribute not set description: {\textstyle 1-p} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d9624b8c566bbbb47130784f9a206a2e4bbc36c8 height: height attribute not set width: width attribute not set description: {\textstyle {\text{or }}1{\text{ in }}(1-p)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b423e434116e0a4774a61858be0d2437b1ec9483 height: height attribute not set width: width attribute not set description: {\textstyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/de61998182ddb364f8b77d67c1aa645685fb3c3b height: height attribute not set width: width attribute not set description: {\displaystyle \phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/57ed565648bb5901c0da2dd3ad10b8d447d4c73c height: height attribute not set width: width attribute not set description: {\displaystyle f^{-1}(p)=\mu +\sigma \phi ^{-1}(p)=\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9f04f02c81a82f110ba3e0bc02b931ddba14c2 height: height attribute not set width: width attribute not set description: {\textstyle \phi ^{-1}(p)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b93ea92fe03865ccbef09efc30804949e5545e height: height attribute not set width: width attribute not set description: {\textstyle z_{p}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee41c1b531933b4fcf571bb55703e37c7a0fc77 height: height attribute not set width: width attribute not set description: {\textstyle \mu +z_{p}\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e909fd422a7987824dffd964868bd1e873ca0d40 height: height attribute not set width: width attribute not set description: {\textstyle \mu \pm z_{p}\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/51e545a85997970e9b6286fdc67de46659468050 height: height attribute not set width: width attribute not set description: {\textstyle 2(1-p)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5207b0523601cf57cae2ee9cb74ce6ac33951595 height: height attribute not set width: width attribute not set description: {\textstyle z_{0.975}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa5d1373db8c4232587a27b948e3cf67aa3d83e height: height attribute not set width: width attribute not set description: {\textstyle \mu \pm 1.96\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b height: height attribute not set width: width attribute not set description: {\textstyle p} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7ca400c2536e7fba7dcda56c02bb014978c01c height: height attribute not set width: width attribute not set description: {\textstyle {\sqrt {2}}\operatorname {erf} ^{-1}(p)=\phi ^{-1}\left({\frac {p+1}{2}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2e59806c8eb5b7858944464c01f370d982d20c72 height: height attribute not set width: width attribute not set description: {\textstyle \phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a982c6635ab3b98d9e12d5f5a8533359bcb38a height: height attribute not set width: width attribute not set description: {\textstyle f(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe66d229cd7ffe26117b0210589335cd0ef9a1a height: height attribute not set width: width attribute not set description: {\textstyle \sigma ^{2}>0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf27b80f77033d27425976fd9d0311e84d8e1a5 height: height attribute not set width: width attribute not set description: {\textstyle x=\mu ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1a67ee38eec71045f0d0821b5043b28dedf4a367 height: height attribute not set width: width attribute not set description: {\textstyle x<\mu ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a9759302b1b0bf87dd4b7d54889c49523c2d19a1 height: height attribute not set width: width attribute not set description: {\textstyle x>\mu ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b983354c35ec0cf9663ad767488067e058ad05b6 height: height attribute not set width: width attribute not set description: {\textstyle x=\mu .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2e0b863fdbf85250f7b351f549f947e75640e7 height: height attribute not set width: width attribute not set description: {\textstyle f'(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4788f1951bbfffb2cce335f34bc1840f8511d682 height: height attribute not set width: width attribute not set description: {\textstyle f''(x)={\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{4}}}f(x).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0d6acba18449501d4f6adb4859fdfd9476f9cb height: height attribute not set width: width attribute not set description: {\textstyle x=\mu -\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd2b2cd6a14795f19ba9c7915c30e69b94d9d82 height: height attribute not set width: width attribute not set description: {\textstyle x=\mu +\sigma .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f84ca2bfc32e1664f86cae4803d641d22cfe08ca height: height attribute not set width: width attribute not set description: {\textstyle \sigma =1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4cd1339f8b2cd458c9821f66a65976c8758cbd height: height attribute not set width: width attribute not set description: {\textstyle \varphi '(x)=-x\varphi (x).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2d52f5d8163787a1bea4e29cd3f6e75d21fe74 height: height attribute not set width: width attribute not set description: {\textstyle \varphi ''(x)=(x^{2}-1)\varphi (x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/de21e66600259ea4b3fa89aa14976b99531fb73b height: height attribute not set width: width attribute not set description: {\textstyle \varphi ^{(n)}(x)=(-1)^{n}\operatorname {he} _{n}(x)\varphi (x),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9b889c7ee6b57ab5b28b71d58bb97631c71f25d6 height: height attribute not set width: width attribute not set description: {\textstyle \operatorname {he} _{n}(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c37706b810427d7356289f56e79feac9da48f931 height: height attribute not set width: width attribute not set description: {\textstyle x^{p}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/84cb2a490da4ae1e7295a3d64f8965cd564dbfda height: height attribute not set width: width attribute not set description: {\textstyle |x|^{p}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e95711ee5bdcd24c18cc65e060a5e9aaa605d3cd height: height attribute not set width: width attribute not set description: {\textstyle \ p} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d2c92b62ac2bbe07a8e475faac29c8cc5f7755 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {e} \left[(x-\mu )^{p}\right]={\begin{cases}0&{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}(p-1)!!&{\text{if }}p{\text{ is even.}}\end{cases}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6479cdc6d9299bee6791fd78541a9caced147e84 height: height attribute not set width: width attribute not set description: {\textstyle n!!} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6788de1792d07b05bdece3a07e26e3ca35cf0dcc height: height attribute not set width: width attribute not set description: {\textstyle n.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dd670f60ea60d0f801a51d07153211c4498c2b9b height: height attribute not set width: width attribute not set description: {\textstyle p,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3b196371c491676efa7ea7770ef56773db7652cd height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\operatorname {e} \left[|x-\mu |^{p}\right]&=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {\frac {2}{\pi }}}&{\text{if }}p{\text{ is odd}}\\1&{\text{if }}p{\text{ is even}}\end{cases}}\\&=\sigma ^{p}\cdot {\frac {2^{p/2}\gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/60bec6c456d39575d4d0bda8ef6126df6965184b height: height attribute not set width: width attribute not set description: {\textstyle p>-1.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dd40f78414e9c6ae54a809ff88f32fb619d8b65d height: height attribute not set width: width attribute not set description: {\textstyle \mu \neq 0,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9b60e0e022990cc8edbdf635ade1ab912951bf32 height: height attribute not set width: width attribute not set description: {\textstyle {}_{1}f_{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fa070da3b4046164d9d5e470611c5372c15c53e5 height: height attribute not set width: width attribute not set description: {\textstyle u.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c17bf881593b86e728bf5dfbdb41a4b86da3875 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\operatorname {e} \left[x^{p}\right]&=\sigma ^{p}\cdot (-i{\sqrt {2}})^{p}u\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right),\\\operatorname {e} \left[|x|^{p}\right]&=\sigma ^{p}\cdot 2^{p/2}{\frac {\gamma \left({\frac {1+p}{2}}\right)}{\sqrt {\pi }}}{}_{1}f_{1}\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right).\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f408e516b4d3056afd68301d502498822d97b9be height: height attribute not set width: width attribute not set description: {\textstyle 0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/971693b90ffed5dde7353deb1fd5537370ae4e60 height: height attribute not set width: width attribute not set description: {\textstyle \mu ^{2}+\sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d39914abc8dfe4369957154c07d84ab2b7a0be9a height: height attribute not set width: width attribute not set description: {\textstyle \mu ^{3}+3\mu \sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/281f395917245069edb45d992b595028b2025234 height: height attribute not set width: width attribute not set description: {\textstyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/94da510d3acb6bee4608e54c4df02c0e6ff805ac height: height attribute not set width: width attribute not set description: {\textstyle 3\sigma ^{4}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/96f63a06184573150760c24d3211e3db3c67f580 height: height attribute not set width: width attribute not set description: {\textstyle \mu ^{5}+10\mu ^{3}\sigma ^{2}+15\mu \sigma ^{4}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1dfa4fb40f5ca504be0cb8c72dcd54e4f68cd8fb height: height attribute not set width: width attribute not set description: {\textstyle \mu ^{6}+15\mu ^{4}\sigma ^{2}+45\mu ^{2}\sigma ^{4}+15\sigma ^{6}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7a69d496ac88f61dd1b90dad400b9dd4dbff26df height: height attribute not set width: width attribute not set description: {\textstyle 15\sigma ^{6}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/29fa39e5d1bae184fdc959de0e68d4a29b5eab74 height: height attribute not set width: width attribute not set description: {\textstyle \mu ^{7}+21\mu ^{5}\sigma ^{2}+105\mu ^{3}\sigma ^{4}+105\mu \sigma ^{6}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c363fa19eb978fe3c38c9ccba3a80fb39b2149f2 height: height attribute not set width: width attribute not set description: {\textstyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/11ebc1faf09bf678b1eede84eb0127e31e43bea9 height: height attribute not set width: width attribute not set description: {\textstyle 105\sigma ^{8}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108 height: height attribute not set width: width attribute not set description: {\textstyle [a,b]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ad97cc40960e6d1d4e65f51596c9cd0c9accfdc0 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {e} \left[x\mid a |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8256dae10b9e3abb3592ff608e81c8bc324edce3 height: height attribute not set width: width attribute not set description: {\textstyle f} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/780ce7d0e407bb8380dfb1d1f2997d11255fa9ba height: height attribute not set width: width attribute not set description: {\textstyle b=\infty } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5beb9ad7c227ee89ebda0dc17818b28959ac9b height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {f}}(t)=\int _{-\infty }^{\infty }f(x)e^{-itx}\,dx=e^{-i\mu t}e^{-{\frac {1}{2}}(\sigma t)^{2}}\,,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0f327332497b21a059c479e7b2ce098baa1a7e height: height attribute not set width: width attribute not set description: {\textstyle i} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/06b652a8fa9ef917c8f5c4843bc5c27904398e89 height: height attribute not set width: width attribute not set description: {\textstyle \varphi _{x}(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2cee41fa7c645c5cd513bda031d2a93db5e0ea2f height: height attribute not set width: width attribute not set description: {\textstyle e^{itx}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bc926f90178739fccd01a96c6fa778ab3535d6 height: height attribute not set width: width attribute not set description: {\textstyle t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8b863dd1facca4d0ada922d4420095387ea547 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi _{x}(t)={\hat {f}}(-t)\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ce1d303b270111ba07bf545bbf6ceca8f6e500dd height: height attribute not set width: width attribute not set description: {\textstyle e^{tx}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5930b107fb4328bc04d077d65ce3d2bf1510de height: height attribute not set width: width attribute not set description: {\displaystyle m(t)=\operatorname {e} \left[e^{tx}\right]={\hat {f}}(it)=e^{\mu t}e^{\sigma ^{2}t^{2}/2}\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40 height: height attribute not set width: width attribute not set description: {\displaystyle k} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/929a04a311aa91f40ff282b49d54f7e0815c28d8 height: height attribute not set width: width attribute not set description: {\displaystyle t^{k}/k!} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560 height: height attribute not set width: width attribute not set description: {\displaystyle t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcd54fe6c5cb4afbfcd7bd94c4778d13b8bbc3f height: height attribute not set width: width attribute not set description: {\displaystyle e[x^{k}]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/44d74b9c48715ac4ee80e4cee69b1a551ee37b7e height: height attribute not set width: width attribute not set description: {\displaystyle g(t)=\ln m(t)=\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ab057858ef1f46aa5e20f095bf0cf86322866a height: height attribute not set width: width attribute not set description: {\textstyle x\sim {\mathcal {n}}(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b9b89fdc012005cf1aea0c28982345f34fe12e height: height attribute not set width: width attribute not set description: {\textstyle {\mathcal {a}}f(x)=\sigma ^{2}f'(x)-(x-\mu )f(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/92b7034b359bf5ee4c81c8992fbb9289a1c6d5a3 height: height attribute not set width: width attribute not set description: {\textstyle {\mathcal {f}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/533b737bedccd7496f904d18a4fd39fade40b282 height: height attribute not set width: width attribute not set description: {\textstyle f:\mathbb {r} \to \mathbb {r} {\mbox{ such that }}\mathbb {e} [|f'(x)|]<\infty } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8986a38a8da1a030622cb86dc493b2e7870c35 height: height attribute not set width: width attribute not set description: {\textstyle x\neq \mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fea648cb5bcbc14d79f97b491fdc5989fa8adaf8 height: height attribute not set width: width attribute not set description: {\textstyle x=\mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/49f7d26e49918fff3216198e4b35ab5f9d64180a height: height attribute not set width: width attribute not set description: {\textstyle \sigma ^{2}=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ec578ebbf0a029f13dca70687f072742277a87ea height: height attribute not set width: width attribute not set description: {\textstyle \delta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9ffd950b0d36c7d350f147a1edd434c1a2cf90 height: height attribute not set width: width attribute not set description: {\textstyle f(x)=\delta (x-\mu ).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/afb24ef768d098fffd78261cbf997a0ff88667ac height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\begin{cases}0&{\text{if }}x<\mu \\1&{\text{if }}x\geq \mu \,.\end{cases}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/129fc3f44c225b79aa5515f97f7321102fd329e8 height: height attribute not set width: width attribute not set description: {\displaystyle h(x)=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx\,,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6a92396d2452d9a8a76d9e66eadf0bd18e6f9600 height: height attribute not set width: width attribute not set description: {\textstyle f(x)\log f(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a69999a3446a1c579f7dcf6aef723ee741c42b8a height: height attribute not set width: width attribute not set description: {\textstyle f(x)=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/856af00c1ea9a107166c797aeb451a2978d699c3 height: height attribute not set width: width attribute not set description: {\displaystyle l=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx-\lambda _{0}\left(1-\int _{-\infty }^{\infty }f(x)\,dx\right)-\lambda _{1}\left(\mu -\int _{-\infty }^{\infty }f(x)x\,dx\right)-\lambda _{2}\left(\sigma ^{2}-\int _{-\infty }^{\infty }f(x)(x-\mu )^{2}\,dx\right)\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/83e87224bdd1053bd82900662d625caf94321aeb height: height attribute not set width: width attribute not set description: {\textstyle \delta f(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd7ec66a3e71534556db59909a49ee04982c706 height: height attribute not set width: width attribute not set description: {\textstyle \delta l} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb88de7e4d31737dae8f02575033272f29e6720 height: height attribute not set width: width attribute not set description: {\textstyle l} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/91b0159db4e0722a4b5ededf3f98d97437605782 height: height attribute not set width: width attribute not set description: {\displaystyle 0=\delta l=\int _{-\infty }^{\infty }\delta f(x)\left(-\ln f(x)-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,dx\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9a80908345c8c165230ad4e818f29470b33064d5 height: height attribute not set width: width attribute not set description: {\displaystyle f(x)=\exp \left(-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/deed531c8e299fbe2e4faf72d198dd0f60661a91 height: height attribute not set width: width attribute not set description: {\textstyle \lambda _{0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/102c20e3f668b3d04e4b0d82f53ff8a1440b5204 height: height attribute not set width: width attribute not set description: {\textstyle \lambda _{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e76660435cf7dc7d47938e1cc36db27a12bafcef height: height attribute not set width: width attribute not set description: {\textstyle \lambda _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2b33217663b3d5e94daf88d51b5de6d3c9c8e6 height: height attribute not set width: width attribute not set description: {\textstyle x\sim n(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c2448beacbfebb0a9ccc54a4927aaa5dde946e77 height: height attribute not set width: width attribute not set description: {\displaystyle h(x)={\tfrac {1}{2}}(1+\ln 2\sigma ^{2}\pi )\,,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c6dea199efa8c774e513875b77ba51bff03b054e height: height attribute not set width: width attribute not set description: {\textstyle \phi _{x}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/354a1bdc69b6b27a0bf4cfaffb65219279017ff6 height: height attribute not set width: width attribute not set description: {\textstyle \phi _{x}(t)=\exp q(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6236f45c1edf052e5a044de03973e4f149bda2 height: height attribute not set width: width attribute not set description: {\textstyle q(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6222becc4f0c5effa012e5335b170575fdbbaad3 height: height attribute not set width: width attribute not set description: {\textstyle y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/049151b05395320c984754f2ded5834c011e90d8 height: height attribute not set width: width attribute not set description: {\textstyle x_{1}\sim n(\mu _{1},\sigma _{1}^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c3719a5439a1563f5a624b1af2dc278963269a height: height attribute not set width: width attribute not set description: {\textstyle x_{2}\sim n(\mu _{2},\sigma _{2}^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/89a5fe7e76e6b9fbe30e34710d2d76d3073d89a6 height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mathrm {kl} }(x_{1}\parallel x_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9f888884d0dcbcbb5ff916c32e1d2bad752517 height: height attribute not set width: width attribute not set description: {\displaystyle h^{2}(x_{1},x_{2})=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\exp \left(-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e882ba24b6d046a40e779c6154532c352ce59f35 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {i}}(\mu ,\sigma ^{2})={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&0\\0&{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9531d09966e9ceeb357705fc047d0c907d3841 height: height attribute not set width: width attribute not set description: {\textstyle x_{1},\ldots ,x_{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfb87c9b047ccf23ace2139d97810dff1ed6670 height: height attribute not set width: width attribute not set description: {\textstyle e[\max _{i}x_{i}]\leq \sigma {\sqrt {2\ln n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aba7a4913188d86c79a9aec22eea9f368b97a692 height: height attribute not set width: width attribute not set description: {\textstyle \mu =-2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8c183fc28435126a9a48808eafb250eedf1a1aa5 height: height attribute not set width: width attribute not set description: {\textstyle \sigma =3} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/767fbd8aca36503020552927a1b915fce17ebf4c height: height attribute not set width: width attribute not set description: {\textstyle x^{y}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4c34bc4eb4a278011a40ffb9f032c80c4cf883 height: height attribute not set width: width attribute not set description: {\textstyle \mu _{y}=2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/08f3ecbda8bf4e133e1cea84384d18699fc2efb7 height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{x}=0.1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7d28eec1e4f61f5b1f4c43952afd9986cd861b53 height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{y}=0.2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2746cd114ffb76f51e39e15f78e568c79eb4d2da height: height attribute not set width: width attribute not set description: {\textstyle \rho _{xy}=0.8} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7cd50265adfa183f69f449014a8f82af9e52ab height: height attribute not set width: width attribute not set description: {\textstyle \mu _{x}=-2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ee10804cc54e77a3fae61edae59f4d0174bdc5ed height: height attribute not set width: width attribute not set description: {\textstyle \mu _{y}=5} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f027fd45ef07328730cafe1a2f98e9220b4bb360 height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{x}^{2}=10} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/72e34e95cd5b78c3141a53554e972ef8f7e3fd8f height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{y}^{2}=20} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/04629b53bb6deeededd41584e7e320bc6deb24f0 height: height attribute not set width: width attribute not set description: {\textstyle \sum _{i=1}^{4}\vert x_{i}\vert } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f1f82180a1f50195618ecb831f661c486af582 height: height attribute not set width: width attribute not set description: {\textstyle ax+b} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2 height: height attribute not set width: width attribute not set description: {\textstyle b} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f90bd224eb54f08933bf7667b6421b083a486397 height: height attribute not set width: width attribute not set description: {\textstyle a\mu +b} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8bc03058160fe2b7e327770351a0d11b794598 height: height attribute not set width: width attribute not set description: {\textstyle e^{x}\sim \ln(n(\mu ,\sigma ^{2}))} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6d34b5937e5ea104e5c93ebf6eb51d8a15145d height: height attribute not set width: width attribute not set description: {\textstyle \sigma (x)\sim p({\mathcal {n}}(\mu ,\,\sigma ^{2}))} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/57ba2b6f77f5ee79ce533737750f805e384e535e height: height attribute not set width: width attribute not set description: {\textstyle {\left|x\right|\sim n_{f}(\mu ,\sigma ^{2})}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5a2a492a37c76731bf40cc0d66591b43faf959 height: height attribute not set width: width attribute not set description: {\textstyle |x-\mu |/\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cfeadb6a15473018854ce00a5f489315858a45a0 height: height attribute not set width: width attribute not set description: {\textstyle |x-\mu |/\sigma \sim \chi _{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2eb302ce8a66dd035d4e07945c03b38a41ee3c height: height attribute not set width: width attribute not set description: {\textstyle x/\sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b3d087928e23e018e2e2e7ff3badaf23bdb536 height: height attribute not set width: width attribute not set description: {\textstyle x^{2}/\sigma ^{2}\sim \chi _{1}^{2}(\mu ^{2}/\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/de1b937be2248646d42f58a296f05ec46eab9ded height: height attribute not set width: width attribute not set description: {\displaystyle \ln p(x)=-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}-\ln \left(\sigma {\sqrt {2\pi }}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ba437038caf7d8aa28caf3bcc915dbca10aa7387 height: height attribute not set width: width attribute not set description: {\textstyle (x-\mu )^{-2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad6c6a48f23684f563c6d8467744a80339c6bdc height: height attribute not set width: width attribute not set description: {\textstyle \sigma ^{-2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8988aef95eb5600d6730ece0631d654408f194d6 height: height attribute not set width: width attribute not set description: {\textstyle x_{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0672d6a80563b72164d70c7c6a0f39f093207de3 height: height attribute not set width: width attribute not set description: {\textstyle \mu _{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/797d1f85ae11f23755ed4bf3d1a1c574911cff40 height: height attribute not set width: width attribute not set description: {\textstyle \mu _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/259e8a5c676195775647fe765266c0a74eeed92b height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{2}^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77ad5dae54aa9dd27f842a7d6dd199386e8b0a0d height: height attribute not set width: width attribute not set description: {\textstyle \mu _{1}+\mu _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/551b9e3d182ae052da11927e652710504685a917 height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{1}^{2}+\sigma _{2}^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fd96b99d81cffac6dcdd636ede2372218fffac12 height: height attribute not set width: width attribute not set description: {\textstyle x+y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/797cd2ee57ec14065d7bdbcc702699d7954c14a0 height: height attribute not set width: width attribute not set description: {\textstyle x-y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/187e0ddb488bc4b83589c8c3b0853bf00b8d80eb height: height attribute not set width: width attribute not set description: {\textstyle 2\sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cecad53efc9fb1f034f57b6ca0dd5754f504c919 height: height attribute not set width: width attribute not set description: {\displaystyle x_{3}={\frac {ax_{1}+bx_{2}-(a+b)\mu }{\sqrt {a^{2}+b^{2}}}}+\mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/873c52c80dce07adc3c2a81eadf14d54f50589bf height: height attribute not set width: width attribute not set description: {\textstyle \alpha =2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e0afe402f1fe01bbec5615f009ecbb2a33001476 height: height attribute not set width: width attribute not set description: {\textstyle x_{k}\sim {\mathcal {n}}(m_{k},\sigma _{k}^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/17da80cc83f4f41abbb5cd2318c8450e76b8efa8 height: height attribute not set width: width attribute not set description: {\textstyle k\in \{0,1\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1156cfcef03aabda6ebc0c9b38eb7a22f16c6b21 height: height attribute not set width: width attribute not set description: {\textstyle {\frac {1}{\int _{\mathbb {r} ^{n}}x_{0}^{\alpha }(x)x_{1}^{1-\alpha }(x)\,{\text{d}}x}}x_{0}^{\alpha }x_{1}^{1-\alpha }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ef0b16243d84fd268ff7749996b279bb6826b1 height: height attribute not set width: width attribute not set description: {\textstyle {\mathcal {n}}(m_{\alpha },\sigma _{\alpha }^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0e5ed5af81505ab74dfef47c0d5d6cd2d0c7de64 height: height attribute not set width: width attribute not set description: {\textstyle m_{\alpha }={\frac {\alpha m_{0}\sigma _{1}^{2}+(1-\alpha )m_{1}\sigma _{0}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5dc27898fde9a32d74f281cb31ecbcbcf4f53c height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{\alpha }^{2}={\frac {\sigma _{0}^{2}\sigma _{1}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9a5ad2791ebcafa159cfbbfe3f678f71c0aea2 height: height attribute not set width: width attribute not set description: {\textstyle x_{1}\pm x_{2}\sim {\mathcal {n}}(0,2)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dd03c2a1da26df0923943cd51f8cca2f7e54a501 height: height attribute not set width: width attribute not set description: {\textstyle z=x_{1}x_{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/059cb792b7e121433075f7b2b303da36d56b8fd0 height: height attribute not set width: width attribute not set description: {\textstyle f_{z}(z)=\pi ^{-1}k_{0}(|z|)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b36f6f66ce8a1a22876ffb7424773bed7e05be height: height attribute not set width: width attribute not set description: {\textstyle k_{0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9272820f880050f29e264a3954ce4b663e57f5a4 height: height attribute not set width: width attribute not set description: {\textstyle \phi _{z}(t)=(1+t^{2})^{-1/2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/88defb0a13fa86397485ffa2f250fe7785d0ca77 height: height attribute not set width: width attribute not set description: {\textstyle x_{1}/x_{2}\sim \operatorname {cauchy} (0,1)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/033c891113a195a08977779d3406751acc19fbda height: height attribute not set width: width attribute not set description: {\textstyle x_{1},x_{2},\ldots ,x_{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6b76ecc4420156e1818fea4a8500786c0cc41ecd height: height attribute not set width: width attribute not set description: {\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}\sim \chi _{n}^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/960c88fa1831b7505d9672de66058532fa5d4053 height: height attribute not set width: width attribute not set description: {\textstyle n-1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/36ff0d3c79a0504e8f259ef99192b825357914d7 height: height attribute not set width: width attribute not set description: {\displaystyle t={\frac {{\overline {x}}-\mu }{s/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(x_{1}+\cdots +x_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left[(x_{1}-{\overline {x}})^{2}+\cdots +(x_{n}-{\overline {x}})^{2}\right]}}}\sim t_{n-1}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/36fb033137341cdfab5dfc6e7aef7f78f87ca631 height: height attribute not set width: width attribute not set description: {\textstyle y_{1},y_{2},\ldots ,y_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1b5d2ab40c3e85b5f24d5b13e8f95202fdca93 height: height attribute not set width: width attribute not set description: {\displaystyle f={\frac {\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)/n}{\left(y_{1}^{2}+y_{2}^{2}+\cdots +y_{m}^{2}\right)/m}}\sim f_{n,m}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0ec175c34bf70d7336eb9f25dbc2d269ff701 height: height attribute not set width: width attribute not set description: {\textstyle q=\sum x_{i}^{2}+\sum x_{j}+c} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d633459065fb34d9100f85db9af130f4ddfe1240 height: height attribute not set width: width attribute not set description: {\textstyle {\text{n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c57b6786aa4c5bd9599caffb5ae4480a3286961e height: height attribute not set width: width attribute not set description: {\textstyle {\frac {\mu }{n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5e51eb9306d3daff888093e26f3e17267ce1fe3d height: height attribute not set width: width attribute not set description: {\textstyle {\frac {\sigma ^{2}}{n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b3dc666b11bdb962170ace940a297ff8d9c7f height: height attribute not set width: width attribute not set description: {\textstyle \sum {a_{k}x_{k}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/701fc598b855acd656fb94b99cfc1696dd881016 height: height attribute not set width: width attribute not set description: {\textstyle \sum {b_{k}x_{k}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8782c28c517e623ccb7715a9e66a964f49446069 height: height attribute not set width: width attribute not set description: {\textstyle x_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4cca7bca2de00b5be911848276eca1c52c8ce765 height: height attribute not set width: width attribute not set description: {\textstyle \sum {a_{k}b_{k}\sigma _{k}^{2}=0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/55cef67d203b726b4d716826adbb3c680d991da5 height: height attribute not set width: width attribute not set description: {\textstyle \sigma _{k}^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a6eb9de0278ee18288eb1a3d7cbe2f397a30ce29 height: height attribute not set width: width attribute not set description: {\displaystyle f_{x}(x)=n(\mu ,\sigma _{1}^{2}){\text{ if }}x\leq \mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/443ae32534241eddf5e4f7b29def2bc2373e2c99 height: height attribute not set width: width attribute not set description: {\displaystyle f_{x}(x)=n(\mu ,\sigma _{2}^{2}){\text{ if }}x\geq \mu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0a8174e5833e49ca75d2a0c411fd29dd17982d height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {e} (x)=\mu +{\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4a8615b5145dfc91cd114e8fb961d90ccecbda height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {v} (x)=\left(1-{\frac {2}{\pi }}\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9959f2c5186e2ed76884054edaf837a602ac6fac height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {t} (x)={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}\right]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/55e09ee050ccb93f44cef510332f40a3d6bc651d height: height attribute not set width: width attribute not set description: {\textstyle (x_{1},\ldots ,x_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/003faa08d27475dd2b029e9f7f0cebab17c0e147 height: height attribute not set width: width attribute not set description: {\displaystyle \ln {\mathcal {l}}(\mu ,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i}\mid \mu ,\sigma ^{2})=-{\frac {n}{2}}\ln(2\pi )-{\frac {n}{2}}\ln \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0269b28e095780b5f1f76c94505841fbe51aeec2 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\mu }}={\overline {x}}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f2946f476cb2518422bed7d85a33eb8d8460d365 height: height attribute not set width: width attribute not set description: {\textstyle \ln {\mathcal {l}}({\hat {\mu }},{\hat {\sigma }}^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/561353e6bc80d226fddd9510be61d21bc67b3aee height: height attribute not set width: width attribute not set description: {\displaystyle \ln {\mathcal {l}}({\hat {\mu }},{\hat {\sigma }}^{2})=(-n/2)[\ln(2\pi {\hat {\sigma }}^{2})+1]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4 height: height attribute not set width: width attribute not set description: {\displaystyle \textstyle {\hat {\mu }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c74eb776989f75b04948837080faa9ebc08c8cd3 height: height attribute not set width: width attribute not set description: {\displaystyle \textstyle {\overline {x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f1fbb023c73b0f4010814107ac36419b16a226 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\mu }}\sim {\mathcal {n}}(\mu ,\sigma ^{2}/n).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/98cf99dd702e8c61031251ee2506b639a6eff98f height: height attribute not set width: width attribute not set description: {\displaystyle \textstyle {\mathcal {i}}^{-1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd0024843448c587ee8246c08fe5af7fb03cc95 height: height attribute not set width: width attribute not set description: {\displaystyle \textstyle 1/{\sqrt {n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/680784f1a8c2242d7a04788c43a18d276b993466 height: height attribute not set width: width attribute not set description: {\textstyle n\rightarrow \infty } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe762c7215a7ac297a3bd441e237f92cd415c00 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {n}}({\hat {\mu }}-\mu )\,\xrightarrow {d} \,{\mathcal {n}}(0,\sigma ^{2}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084 height: height attribute not set width: width attribute not set description: {\displaystyle \textstyle {\hat {\sigma }}^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/24ac5be3cbea7265af53b722b53ab3077320e713 height: height attribute not set width: width attribute not set description: {\textstyle s^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/45013da7f502d373d039cb1056a9c4d1ea06ffc7 height: height attribute not set width: width attribute not set description: {\textstyle s} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bb09766b1fa03887c9ec7f7254e3b25f94224532 height: height attribute not set width: width attribute not set description: {\displaystyle s^{2}={\frac {n}{n-1}}{\hat {\sigma }}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b55e6d2c748d5ba1ff42692650492b9506ab164d height: height attribute not set width: width attribute not set description: {\displaystyle s^{2}\sim {\frac {\sigma ^{2}}{n-1}}\cdot \chi _{n-1}^{2},\qquad {\hat {\sigma }}^{2}\sim {\frac {\sigma ^{2}}{n}}\cdot \chi _{n-1}^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bb270a41c1ca6bb65b250cab50e47b302aca984f height: height attribute not set width: width attribute not set description: {\textstyle 2\sigma ^{4}/(n-1)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/64e1884a9be0b16cd8d2bfbe88f08e7ca6b02a45 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {n}}({\hat {\sigma }}^{2}-\sigma ^{2})\simeq {\sqrt {n}}(s^{2}-\sigma ^{2})\,\xrightarrow {d} \,{\mathcal {n}}(0,2\sigma ^{4}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77e278d37f09ceed447765fd8f45c5d9e376aea9 height: height attribute not set width: width attribute not set description: {\displaystyle t={\frac {{\hat {\mu }}-\mu }{s^{2}/{\sqrt {n}}}}={\frac {{\overline {x}}-\mu }{\sqrt {{\frac {1}{n(n-1)}}\sum (x_{i}-{\overline {x}})^{2}}}}\sim t_{n-1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0e735bcff28ced03579c1389a795957b85b606 height: height attribute not set width: width attribute not set description: {\displaystyle \mu \in \left[{\hat {\mu }}-{\frac {t_{n-1,1-\alpha /2}}{\sqrt {n}}}s^{2},\,{\hat {\mu }}+{\frac {t_{n-1,1-\alpha /2}}{\sqrt {n}}}s^{2}\right],} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8fc65b99199a5894b71f3089dc1643322dd875 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{2}\in \left[{\frac {n-1}{\chi _{n-1,1-\alpha /2}^{2}}}s^{2},\,{\frac {n-1}{\chi _{n-1,\alpha /2}^{2}}}s^{2}\right],} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/be9788127400ca78b183ca3f1a956bdda80ef5e9 height: height attribute not set width: width attribute not set description: {\displaystyle \mu \in \left[{\hat {\mu }}-{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s^{2},\,{\hat {\mu }}+{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s^{2}\right],} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a019ba33d7f994b3e3bf95f65a69db767894a3cb height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{2}\in \left[s^{2}-{\sqrt {2}}{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s^{2},\,s^{2}+{\sqrt {2}}{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s^{2}\right],} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/11fad9789c407013d2c4cb224bfa84b320563fd9 height: height attribute not set width: width attribute not set description: {\textstyle \textstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dfac4114765b1f994800c9b424b82564b57ba179 height: height attribute not set width: width attribute not set description: {\displaystyle a(x-y)^{2}+b(x-z)^{2}=(a+b)\left(x-{\frac {ay+bz}{a+b}}\right)^{2}+{\frac {ab}{a+b}}(y-z)^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b27b947e13b22bb972ea8cc460c5ab3de2db0237 height: height attribute not set width: width attribute not set description: {\textstyle {\frac {ay+bz}{a+b}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/042395d4562e6a427ab04e879ba859a10e8087a7 height: height attribute not set width: width attribute not set description: {\textstyle {\frac {ab}{a+b}}={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7021bc974153f898ddadd50b0351d9f3a28de8a8 height: height attribute not set width: width attribute not set description: {\textstyle {\frac {ab}{a+b}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9889111e6949ae57adb3a883df2f8a29bb5062 height: height attribute not set width: width attribute not set description: {\textstyle k\times k} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6374f98fcb11f7c1273b06c44e1c0f0b84154048 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}&(\mathbf {y} -\mathbf {x} )'\mathbf {a} (\mathbf {y} -\mathbf {x} )+(\mathbf {x} -\mathbf {z} )'\mathbf {b} (\mathbf {x} -\mathbf {z} )\\={}&(\mathbf {x} -\mathbf {c} )'(\mathbf {a} +\mathbf {b} )(\mathbf {x} -\mathbf {c} )+(\mathbf {y} -\mathbf {z} )'(\mathbf {a} ^{-1}+\mathbf {b} ^{-1})^{-1}(\mathbf {y} -\mathbf {z} )\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/267a22091cc9d9afb86fcacebcc6b842cb0e9b1b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {c} =(\mathbf {a} +\mathbf {b} )^{-1}(\mathbf {a} \mathbf {y} +\mathbf {b} \mathbf {z} )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef06ff3139875b96fe704a43bbecebacdbea460 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} '\mathbf {a} \mathbf {x} =\sum _{i,j}a_{ij}x_{i}x_{j}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e9baf61d13d13e6695b57c3f31856e72c860f8 height: height attribute not set width: width attribute not set description: {\textstyle x_{i}x_{j}=x_{j}x_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c48f5fdcc3f2d80c5a75c3714a154a7e16a3195f height: height attribute not set width: width attribute not set description: {\textstyle a_{ij}+a_{ji}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8d25486eab8a57da216fb9418eb8b4fa889c7b03 height: height attribute not set width: width attribute not set description: {\textstyle \mathbf {x} '\mathbf {a} \mathbf {y} =\mathbf {y} '\mathbf {a} \mathbf {x} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6abcabe83cd01aabf39c16b0bc67994086519d02 height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{i=1}^{n}(x_{i}-\mu )^{2}=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/17005337073440fa8d7c41536f875c6cd5d1fc0e height: height attribute not set width: width attribute not set description: {\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bf563c08aa87170a438239b0d291a4093fd2cb27 height: height attribute not set width: width attribute not set description: {\textstyle x\sim {\mathcal {n}}(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/83dac9d557f3dae598f1438c0a8164d82501fe19 height: height attribute not set width: width attribute not set description: {\textstyle x\sim {\mathcal {n}}(\mu ,1/\tau )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2f12cd6980bf93122c047ab32474d8403317756b height: height attribute not set width: width attribute not set description: {\textstyle \mu \sim {\mathcal {n}}(\mu _{0},1/\tau _{0}),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c2bcd1c34520a24e29b758a0f7427e79e9d8a414 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mathbf {x} \mid \mu ,\tau )&=\prod _{i=1}^{n}{\sqrt {\frac {\tau }{2\pi }}}\exp \left(-{\frac {1}{2}}\tau (x_{i}-\mu )^{2}\right)\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left(-{\frac {1}{2}}\tau \sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right].\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/96e309ead00fbc8603eced5342aa5df534522d6a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mu \mid \mathbf {x} )&\propto p(\mathbf {x} \mid \mu )p(\mu )\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]{\sqrt {\frac {\tau _{0}}{2\pi }}}\exp \left(-{\frac {1}{2}}\tau _{0}(\mu -\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(n\tau ({\bar {x}}-\mu )^{2}+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&=\exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar {x}}-\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}\right)\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b92a63cba1c7539f1d484ee0296c910912780a79 height: height attribute not set width: width attribute not set description: {\textstyle {\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/23717a4ce103170f3da265af44ce96e80461871e height: height attribute not set width: width attribute not set description: {\textstyle n\tau +\tau _{0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a45b361f59d044be9a7d87bf92514795f38419c8 height: height attribute not set width: width attribute not set description: {\displaystyle p(\mu \mid \mathbf {x} )\sim {\mathcal {n}}\left({\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}},{\frac {1}{n\tau +\tau _{0}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cfbdf504b1a9ce4cbe79561b4ae983fdf7271d height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\tau _{0}'&=\tau _{0}+n\tau \\[5pt]\mu _{0}'&={\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\\[5pt]{\bar {x}}&={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/66ef5983f162d4e49610bc8240e713cc2bbca7d8 height: height attribute not set width: width attribute not set description: {\textstyle {\bar {x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ea454c8840683777ce8192d9ae63068c63962858 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\sigma _{0}^{2}}'&={\frac {1}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]\mu _{0}'&={\frac {{\frac {n{\bar {x}}}{\sigma ^{2}}}+{\frac {\mu _{0}}{\sigma _{0}^{2}}}}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]{\bar {x}}&={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2528fe4774a93087d4adae570ef9ab84707f52 height: height attribute not set width: width attribute not set description: {\displaystyle p(\sigma ^{2}\mid \nu _{0},\sigma _{0}^{2})={\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\nu _{0}/2}}{\gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\propto {\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cc06aa31588bba03e4748f8f345f0638a75dc156 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mathbf {x} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right]\\&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {s}{2\sigma ^{2}}}\right]\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/56adf28a77173ce852c7de7eeee102b2f6895b39 height: height attribute not set width: width attribute not set description: {\displaystyle s=\sum _{i=1}^{n}(x_{i}-\mu )^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/381c1b93f6dc76e2cdca9f3f1f77132dd51dc55f height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\sigma ^{2}\mid \mathbf {x} )&\propto p(\mathbf {x} \mid \sigma ^{2})p(\sigma ^{2})\\&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {s}{2\sigma ^{2}}}\right]{\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\frac {\nu _{0}}{2}}}{\gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\\&\propto \left({\frac {1}{\sigma ^{2}}}\right)^{n/2}{\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\exp \left[-{\frac {s}{2\sigma ^{2}}}+{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]\\&={\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}+n}{2}}}}}\exp \left[-{\frac {\nu _{0}\sigma _{0}^{2}+s}{2\sigma ^{2}}}\right]\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d9cea4f20a8750894be82fb32d617284c433fd height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\nu _{0}'&=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/192be53c5d9d249b2ef7ca5622430b689f1aee64 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\nu _{0}'&=\nu _{0}+n\\{\sigma _{0}^{2}}'&={\frac {\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{\nu _{0}+n}}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6242673d0e1932e640fa7ebb2167edbb20535f35 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\alpha '&=\alpha +{\frac {n}{2}}\\\beta '&=\beta +{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{2}}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bab8dee515d3208f73dd85d1cb46706e3a9097f9 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&\sim {\mathcal {n}}(\mu _{0},\sigma ^{2}/n_{0})\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&\sim i\chi ^{2}(\nu _{0},\sigma _{0}^{2})=ig(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/673b045d8322e2ce9e1ecc33c00585873b85547a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\bar {x}}&={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\\\mu _{0}'&={\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\\n_{0}'&=n_{0}+n\\\nu _{0}'&=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/97dcd132cd10175d6ce232772d0c5c5964a9f195 height: height attribute not set width: width attribute not set description: {\textstyle \nu _{0}'{\sigma _{0}^{2}}'} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7afb8e3b63fb1526171840344b32458e55cf8b height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&\sim {\mathcal {n}}(\mu _{0},\sigma ^{2}/n_{0})={\frac {1}{\sqrt {2\pi {\frac {\sigma ^{2}}{n_{0}}}}}}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\&\propto (\sigma ^{2})^{-1/2}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&\sim i\chi ^{2}(\nu _{0},\sigma _{0}^{2})=ig(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\\&={\frac {(\sigma _{0}^{2}\nu _{0}/2)^{\nu _{0}/2}}{\gamma (\nu _{0}/2)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+\nu _{0}/2}}}\\&\propto {(\sigma ^{2})^{-(1+\nu _{0}/2)}}\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right].\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f808161077baef3854dbfd90b870698d721090 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2};\mu _{0},n_{0},\nu _{0},\sigma _{0}^{2})&=p(\mu \mid \sigma ^{2};\mu _{0},n_{0})\,p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})\\&\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right].\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d77342aadcb34c5d84418cecaefdb52842b6b7 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mathbf {x} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\right]\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/29b915f070b522a1e9f419be05624c86c854ca14 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mathbf {x} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&\propto {\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(s+n({\bar {x}}-\mu )^{2}\right)\right]\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/02a32ce18746fffb539a3316846c9a6b40695c31 height: height attribute not set width: width attribute not set description: {\textstyle s=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cad9489034d77d53c12c7ee6044f712cfdb77831 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2}\mid \mathbf {x} )&\propto p(\mu ,\sigma ^{2})\,p(\mathbf {x} \mid \mu ,\sigma ^{2})\\&\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right]{\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(s+n({\bar {x}}-\mu )^{2}\right)\right]\\&=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+s+n_{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+s+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}+(n_{0}+n)\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right)\right]\\&\propto (\sigma ^{2})^{-1/2}\exp \left[-{\frac {n_{0}+n}{2\sigma ^{2}}}\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right]\\&\quad \times (\sigma ^{2})^{-(\nu _{0}/2+n/2+1)}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+s+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right]\\&={\mathcal {n}}_{\mu \mid \sigma ^{2}}\left({\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}},{\frac {\sigma ^{2}}{n_{0}+n}}\right)\cdot {\rm {ig}}_{\sigma ^{2}}\left({\frac {1}{2}}(\nu _{0}+n),{\frac {1}{2}}\left(\nu _{0}\sigma _{0}^{2}+s+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right).\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9734bce3c2a0076e7414aadcb56bed5e7c09ec9a height: height attribute not set width: width attribute not set description: {\textstyle p(\mu |\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/23d06552135aaa8a49d59cd28e4fc282a247bdd6 height: height attribute not set width: width attribute not set description: {\textstyle p(\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f0268a17b46d104a4b1ab3967ead4e7da28d4b height: height attribute not set width: width attribute not set description: {\textstyle {\frac {\partial }{\partial t}}f(x,t)={\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}f(x,t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0b78b479cc1c1ccd06f8cdfd31223335921e5a5b height: height attribute not set width: width attribute not set description: {\textstyle g(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/51fa20f18a8a5ed19c147db4686e7b15b6ca2e38 height: height attribute not set width: width attribute not set description: {\displaystyle x={\sqrt {-2\ln u}}\,\cos(2\pi v),\qquad y={\sqrt {-2\ln u}}\,\sin(2\pi v).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bdace1879c7c786ba946a60e5acb29f354d86796 height: height attribute not set width: width attribute not set description: {\displaystyle x=u{\sqrt {\frac {-2\ln s}{s}}},\qquad y=v{\sqrt {\frac {-2\ln s}{s}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/202a295cd562d4d7404a1042e23f14b8d72be308 height: height attribute not set width: width attribute not set description: {\displaystyle \phi (x)=1-\varphi (x)\left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\right)+\varepsilon (x),\qquad t={\frac {1}{1+b_{0}x}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ca45895a9095ca37f734f18a83481576ba4c5a49 height: height attribute not set width: width attribute not set description: {\displaystyle \phi (x)={\frac {1}{2}}+\varphi (x)\left(x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+{\frac {x^{7}}{3\cdot 5\cdot 7}}+{\frac {x^{9}}{3\cdot 5\cdot 7\cdot 9}}+\cdots \right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/80cf15e7b1c8138c3b5cc37f31168f914c7d6621 height: height attribute not set width: width attribute not set description: {\textstyle 1-\phi } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/208228d226c31bceb0c8aeadfab59460f19a157e height: height attribute not set width: width attribute not set description: {\textstyle 2^{-53}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/191b9343ac891e152b2093e8b6ac149e367dbf58 height: height attribute not set width: width attribute not set description: {\textstyle \left(\approx 1.1\times 10^{-16}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c434fc1e2ab777786469de853c75e616007b3eb4 height: height attribute not set width: width attribute not set description: {\textstyle x\geq 0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9a049b86b4971707745b5c6ba2e40ae4e25205 height: height attribute not set width: width attribute not set description: {\textstyle {\begin{aligned}1-\phi \left(x\right)&=\left({\frac {0.39894228040143268}{x+2.92678600515804815}}\right)\left({\frac {x^{2}+8.42742300458043240x+18.38871225773938487}{x^{2}+5.81582518933527391x+8.97280659046817350}}\right)\\&\left({\frac {x^{2}+7.30756258553673541x+18.25323235347346525}{x^{2}+5.70347935898051437x+10.27157061171363079}}\right)\left({\frac {x^{2}+5.66479518878470765x+18.61193318971775795}{x^{2}+5.51862483025707963x+12.72323261907760928}}\right)\\&\left({\frac {x^{2}+4.91396098895240075x+24.14804072812762821}{x^{2}+5.26184239579604207x+16.88639562007936908}}\right)\left({\frac {x^{2}+3.83362947800146179x+11.61511226260603247}{x^{2}+4.92081346632882033x+24.12333774572479110}}\right)e^{-{\frac {x^{2}}{2}}}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/11dbe80785d8f5d86eb8e91c35b6f3003f8d2838 height: height attribute not set width: width attribute not set description: {\textstyle x<0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2e12b409099845b3057fa7bdb2d9b84c6cacf73a height: height attribute not set width: width attribute not set description: {\displaystyle 1-\phi \left(x\right)=1-\left(1-\phi \left(-x\right)\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2df7f1427d0c90d075faef38f4f5ab7acce5c9 height: height attribute not set width: width attribute not set description: {\displaystyle z=\phi ^{-1}(p)=5.5556\left[1-\left({\frac {1-p}{p}}\right)^{0.1186}\right],\qquad p\geq 1/2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e1edea9f990058f741db6735799c8b40999b833b height: height attribute not set width: width attribute not set description: {\displaystyle z=-0.4115\left\{{\frac {1-p}{p}}+\log \left[{\frac {1-p}{p}}\right]-1\right\},\qquad p\geq 1/2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b69fa586cffdfbbd40a94c65629726e4ae78bf height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}l(z)&=\int _{z}^{\infty }(u-z)\varphi (u)\,du=\int _{z}^{\infty }[1-\phi (u)]\,du\\[5pt]l(z)&\approx {\begin{cases}0.4115\left({\dfrac {p}{1-p}}\right)-z,&p<1/2,\\\\0.4115\left({\dfrac {1-p}{p}}\right),&p\geq 1/2.\end{cases}}\\[5pt]{\text{or, equivalently,}}\\l(z)&\approx {\begin{cases}0.4115\left\{1-\log \left[{\frac {p}{1-p}}\right]\right\},&p<1/2,\\\\0.4115{\dfrac {1-p}{p}},&p\geq 1/2.\end{cases}}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bd21c29efa71343458e18c6c3bdd7a1005cafa0d height: height attribute not set width: width attribute not set description: {\textstyle \phi ^{-1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9fee018963a079bb837482314ae6b1533a3a19 height: height attribute not set width: width attribute not set description: {\textstyle 2^{n}/{\sqrt {2\pi n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d5327ed8841e09d62970ee806553294cdfe96e9e height: height attribute not set width: width attribute not set description: {\textstyle -{\frac {2\ell \ell }{n}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/9/9b/carl_friedrich_gauss.jpg/180px-carl_friedrich_gauss.jpg height: 232 width: 180 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c45300f5e3b84f9d3571c95d621dc76c4097b4b3 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi {\mathit {\delta }}={\frac {h}{\surd \pi }}\,e^{-\mathrm {hh} \delta \delta },} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/75ebaa526f97ad5136df9d8a540d1970aa8c5664 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {n} {\frac {1}{\alpha \;{\sqrt {\pi }}}}\;e^{-{\frac {x^{2}}{\alpha ^{2}}}}\,dx} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fcecb090761f24d376d5f8f4962eb0459504fcae height: height attribute not set width: width attribute not set description: {\displaystyle df={\frac {1}{\sqrt {2\sigma ^{2}\pi }}}e^{-(x-m)^{2}/(2\sigma ^{2})}\,dx.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3c3bf34fb8926a1f042fdbf486b618904a66d6 height: height attribute not set width: width attribute not set description: {\textstyle (0,\infty )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/70cd41c754f1c4cc4a6a7349ac38ddc500b149d0 height: height attribute not set width: width attribute not set description: {\textstyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}} |
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