en.wikipedia.org website review
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SEO Keyword summary for en.wikipedia.org/wiki/gamma_distribution
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be gamma
Focus keyword
Short and long tail
Short Tail Keywords gamma displaystyle distribution |
long Tail Keywords (2 words) gamma distribution shape parameter gamma alpha exponential family stable count |
long Tail Keywords (3 words) move to sidebar distribution with parameters shape parameter k stable count distribution distribution with rate if the shape distribution is given |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/gamma_distribution page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
Page title
Title length
gamma distribution wikipedia
Meta description
Meta description legth
Meta description SEO
No meta relevance in the description detected !
Content SEO
Number of Words
Spam detected?
Headings
Heading distribution
Heading normalisation
Heading SEO impact
Emphasis (bold and italic)
Emphasis SEO impact
Images
Number of images
Images dimensions
Image alt descriptions
Images SEO impact
wikipedia free encyclopedia probability density plots gamma distributions cumulative distribution displaystyle xin infty fxfrac ktheta kxk extheta beta alpha xalpha ebeta kgamma leftkfrac xtheta right frac theta text kgeq geq sqrt beginalignedkln kpsi kendaligned beginalignedalpha psi endaligned tktext tfrac left tbeta rightalpha itk itbeta iktheta beginpmatrixpsi endpmatrix ialpha kfrac vxquad quad vxexquad exvx xsim equiv operatorname beginalignedfxalpha xbeta ptendaligned fxalpha int xfualpha dufrac xgamma sum xiiebeta xebeta xsum xii fxktheta kquad xfuktheta rightgamma ileftfrac rightiextheta ikinfty righti sigma mathrm xntheta nfrac kngamma nprod nki ldots kint dxfrac textgammak kkfrac oleftfrac rightcdots kapprox kak egamma knu kke kleq log kkleft kright kabk linfty klog kto kegamma kleftfrac prime approx tilde gknu gkfrac uknu nxisim leftsum nkitheta cxsim kctheta leftalpha cright xpsi kln var beginalignedoperatorname xoperatorname ptoperatorname ptalpha xkln beginaligneddmathrm pbeta palpha qbeta qalpha qpsi plog qlog pfrac pendaligned kptheta pkqtheta qkpkqpsi kplog kqkqlog pkpfrac ptheta qtheta qendaligned skfrac sbeta ixi textstyle bar xfrac nsum ixisim nnlambda kin mathbf qquad ysim pois leftfrac pxxpyk sim asqrt atheta xksim kbeta fxmid expbeta leftbeginmatrixleftalpha endmatrixzright igb fxkk fxkdisplaystyle uwkleftfrac xurightleftkuk mathfrak kleftukrightrightdu nalpha wkx lktheta prod nfxiktheta ell nln xisum xitheta nkln hat knsum nxifrac xinknkln xiknrightnln ngeq nxirightfrac xiln xoverline overline kleft sln ssqrt kleftarrow ksfrac khat beginalignedhat xtextandpsi xendaligned varepsilon ptextunderflow fvarepsilon times ptextunderflowapprox usim textuniform zlnylnuk expzsim textgammaktheta nxinsum nxiln nxisum leftnsum xiright kbar xhat xln xbar nleft krightfrac mid dots xnpropto yequiv nxiqquad xncxitheta eytheta meytheta dtheta xnk mexydxynkmgamma nkm xmfrac nkmgamma nkym ynk pktheta pqrsfrac zfrac etheta qgamma krtheta beginalignedppprod nolimits ixiqqsum ixirrnssnendaligned uksim uleq eedelta delta eta wxi wexi exi leftxi lfloor krfloor uirightsim scriptstyle dafrac cfrac ufrac ddvdln leq aalpha wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/gamma_distribution
Mobile rendering
Mobile optimizations
Responsive design detected (mobile css)
No flash detected !
Mobile improvement
Marketing / lead generation for en.wikipedia.org/wiki/gamma_distribution
Social Media
Facebook shares | Facebook likes | ||
Facebook comments | Tweets | ||
Google +1 |
Conversion form
Search form
Analytics
Online presence
SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
distribution found in path !
gamma found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
Favicon icon found?
Robots.txt found?
Sitemap found?
Navigation and internal links
Navigation
Url seperator
Human readable urls
Number of links
Link SEO Impact
statistics
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en.m.wikipedia.org |
en.wikipedia.org wikimedia foundation inc
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w view history
httpsenwikipediaorgwindexphptitlegammadistributionoldid1241370584
page information
printable version
herman rubin
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wiki read
parameter
shape parameter
scale parameter
rate parameter
support
probability density function
cumulative distribution function
mean
median
mode
variance
skewness
excess kurtosis
information entropy
momentgenerating function
fisher information
method of moments
probability theory
statistics
probability distributions
exponential
erlang
chisquared
econometrics
bayesian statistics
conjugate prior
maximum entropy
digamma function
life testing
random variables
poisson
inverse
normal gaussian
integer
gamma function
incomplete gamma function
coefficient of variation
raw moment
laurent series
ramanujans displaystyle theta function
loglog plot
eulermascheroni constant
wilsonhilferty transformation
closedform expressions
exponential integral
independent
infinite divisibility
exponential
natural parameters
natural statistics
natural exponential
trigamma function
kullbackleibler divergence
laplace transform
schulzzimm distribution
poisson process
maxwellboltzmann
generalized
beta
dirichlet
mean
matrix gamma
wishart
generalized integer gamma distribution
generalized
negative binomial
tweedie
exponential dispersion models
halfnormal
modified halfnormal distribution
foxwright psi function
beta prime
compound distribution
compound gamma distribution
kdistribution
weibull
stable count distribution
maximum likelihood
sample mean
concave
inequality properties of the polygamma function
newtons method
floating point
prior
bayesian inference
pareto
gompertz
celldivision events
atp synthase
insurance claims
gamma process
poisson regression
mixture
multipath fading
rayleigh
rician distribution
oncology
cancer
incidence
driver events
neuroscience
interspike intervals
bacterial
gene expression
copy number
constitutively expressed
protein molecules
genomics
peak calling
chipchip
chipseq
precision
phylogenetics
maximum likelihood
bayesian
distance matrix methods
uniformly distributed
inverse transform sampling
acceptancerejection method
matlab
citeseerx
hogg r v
isbn
blei david m
arxiv
issn
s2cid
plos one
bibcode
pmid
oclc
jstor
encyclopedia of mathematics
ems press
weisstein eric w
mathworld
wayback machine
benford
bernoulli
betabinomial
binomial
categorical
hypergeometric
poisson binomial
rademacher
soliton
discrete uniform
zipf
zipfmandelbrot
beta negative binomial
borel
conwaymaxwellpoisson
discrete phasetype
delaporte
extended negative binomial
floryschulz
gausskuzmin
geometric
logarithmic
mixed poisson
parabolic fractal
skellam
yulesimon
zeta
arcsine
argus
baldingnichols
bates
beta rectangular
continuous bernoulli
irwinhall
kumaraswamy
logitnormal
noncentral beta
pert
raised cosine
reciprocal
triangular
uquadratic
uniform
wigner semicircle
benini
benktander 1st kind
benktander 2nd kind
burr
noncentral
dagum
davis
hyperexponential
hypoexponential
logarithmic
noncentral
folded normal
frchet
gammagompertz
shifted
halflogistic
hotellings tsquared
inverse gaussian
kolmogorov
logcauchy
loglaplace
loglogistic
lognormal
logt
lomax
matrixexponential
maxwelljttner
mittagleffler
nakagami
phasetype
polyweibull
relativistic breitwigner
rice
truncated normal
type2 gumbel
discrete
wilkss lambda
cauchy
generalized normal
fishers z
kaniadakis gaussian
gaussian q
generalized hyperbolic
geometric stable
gumbel
holtsmark
hyperbolic secant
johnsons su
landau
laplace
asymmetric
logistic
noncentral t
normalinverse gaussian
skew normal
slash
stable
students t
tracywidom
variancegamma
voigt
generalized chisquared
generalized extreme value
generalized pareto
marchenkopastur
kaniadakis exponential
kaniadakis gamma
kaniadakis weibull
kaniadakis logistic
kaniadakis erlang
qexponential
qgaussian
qweibull
shifted loglogistic
tukey lambda
rectified gaussian
multivariate joint
multinomial
dirichlet
negative
generalized
multivariate laplace
multivariate normal
multivariate stable
multivariate t
normalgamma
matrixvalued
matrix normal
matrix t
normal
inverse
normalinverse
complex
directional
circular uniform
univariate von mises
wrapped normal
wrapped cauchy
wrapped exponential
wrapped asymmetric laplace
wrapped lvy
kent
bivariate von mises
von misesfisher
bingham
degenerate
singular
dirac delta function
cantor
circular
compound poisson
elliptical
locationscale
pearson
wrapped
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Links to external pages
Outloing links
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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.
Item | Factor | Pointers | |
---|---|---|---|
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 31 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. | |
Meta description relevance | 0% | Meta Description should reflect the contents of a site. This site has a 0 % match | |
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 531 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 3 level 1 folders and 7 folders above or in the first level of navigation. | |
Headings | 48% | Headers should reflect the contents of a site. This site has a 21 % match | |
Links | 8% | Link anchors should to some degree reflect the contents of a site. This site has a 4 % match | |
Image alt tags | 17% | Image alt tags should to some degree reflect the contents of a site. This site has a 6 % match | |
Bold and italic | 6% | Bold and italic tags should reflect the contents of a site to some degree. This site has a 2 % match | |
Html ratio | 30% | Try to keep the html / text ratio as low as possible. More html means longer loading times. Layout should be handled in a serpate css file | |
Image descriptions | 96% | 95.515695067265 % of all images have been described via the "alt" attribute. Describing images with relevant text may lead to better results in the search engines. | |
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 8536 words | |
Server response time | 30% | A slow server slows down a website. This server responds 6.12% slower the average | |
Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
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 585 inline style declarations ( <a style="color:green">) with a size of 17648 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 | |
Flash | 100% | Perfect, we detected no flash objects on your page | |
Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
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). |
How would you like to have SEO advice for all your pages ?? Start your SEO Dashboard and optimize your website!
en.wikipedia.org images and descriptions
210 images found at en.wikipedia.org Images can improve the user experience for a website by making a pag visually appealing Images can also add extra keyword relevance to a webpage by using alt tags. Images can also slow down a website. If the width and height for a picture is not specified for a browser know in advance how large the image is. A browser must first load the picture and see before it knows how much space should be on the page. Upon reservation In the meantime, the browser can do little but wait. When the height and width for the plate are given in the HTML code, a browser just continues to build for a page while the images load in the background.
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/e/e6/gamma_distribution_pdf.svg/325px-gamma_distribution_pdf.svg.png height: 244 width: 325 description: probability density plots of gamma distributions |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/gamma_distribution_cdf.svg/325px-gamma_distribution_cdf.svg.png height: 244 width: 325 description: cumulative distribution plots of gamma distributions |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb9a21cd92ae991a12ba8aaa186dd60922862c0 height: height attribute not set width: width attribute not set description: {\displaystyle x\in (0,\infty )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/959c7540287d0d1be72c2e39826eaf2060ce907e height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\frac {1}{\gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3373bd46a7261833556cec0a5c95f51ab4b0e74d height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\frac {\beta ^{\alpha }}{\gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/206171e583bb2aa2e14a65530b7cd9e5596d922e height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\frac {1}{\gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd8662ff6fdff9e9f96399ac86737471db7d976 height: height attribute not set width: width attribute not set description: {\displaystyle f(x)={\frac {1}{\gamma (\alpha )}}\gamma (\alpha ,\beta x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/128f9dbb916e39f620213e206592bf14f13554d2 height: height attribute not set width: width attribute not set description: {\displaystyle k\theta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7e9485a5ec6938007ea1b5f5f09946f39a14a42f height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\alpha }{\beta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9e723ed5d8686d56eda7a66c3f1a1fe3db880f height: height attribute not set width: width attribute not set description: {\displaystyle (k-1)\theta {\text{ for }}k\geq 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f83ac368070086266d1302f673abb085ec278d46 height: height attribute not set width: width attribute not set description: {\displaystyle 0{\text{ for }}k<1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b0209d0c976e5707524172b0d4071a3a73ad6f12 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9fb845e6076205ac8deabbbc533acedc15103d height: height attribute not set width: width attribute not set description: {\displaystyle k\theta ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/84135b4731462ff6c9a9c23a434a06e9305ee85e height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\alpha }{\beta ^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/45eff79aec609af84bf67d9272aebc1fa8681994 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {2}{\sqrt {k}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8a164ebce39b73870af9d23275a9a76c15e6eb height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {2}{\sqrt {\alpha }}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bea39cf9fc4c3ff5dc63035a85205b437db7e426 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {6}{k}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ce5c506f119628c417e693b0368db51ee874ec height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {6}{\alpha }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/badcd9de56aee1547b1743cd802484471cf3555c height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}k&+\ln \theta +\ln \gamma (k)\\&+(1-k)\psi (k)\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a73a699028dd881daebebc59d36f894e74f187 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/48369eff20942cb3a2e4cfbef6c31d75c2fca94b height: height attribute not set width: width attribute not set description: {\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aae293b4a1441aae3cafacb8a4ce61d23bfa48cb height: height attribute not set width: width attribute not set description: {\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5768a597d0f48f216a027bd3e92a8b3e67e0371b height: height attribute not set width: width attribute not set description: {\displaystyle (1-\theta it)^{-k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9b1c060b013f2c7f698d58711983d9ddf0b18f height: height attribute not set width: width attribute not set description: {\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/96ca1fa0ecbcf09ce7fc348b6c3363faac0408e4 height: height attribute not set width: width attribute not set description: {\displaystyle i(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6881c4895c0e9bbc951e99f996f94e2a3690aae8 height: height attribute not set width: width attribute not set description: {\displaystyle i(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/79060985aa8683bbf0b380d57ca56522822342ca height: height attribute not set width: width attribute not set description: {\displaystyle k={\frac {e[x]^{2}}{v[x]}}\quad \quad } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/de7bf8b64325f4129e05929e2385f3ca37bb88bf height: height attribute not set width: width attribute not set description: {\displaystyle \theta ={\frac {v[x]}{e[x]}}\quad \quad } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/87074b8ec525badd064920b64dcff7be1c51ceaa height: height attribute not set width: width attribute not set description: {\displaystyle \alpha ={\frac {e[x]^{2}}{v[x]}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/187bc571898043026331662ae41bb70d4104d429 height: height attribute not set width: width attribute not set description: {\displaystyle \beta ={\frac {e[x]}{v[x]}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/492bbf1283920d57a534b63fca94a2e2ff4d1946 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha =k} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/664250d7209fe5e66b2de5a0c74c7a28ef17b395 height: height attribute not set width: width attribute not set description: {\displaystyle \beta =1/\theta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3 height: height attribute not set width: width attribute not set description: {\displaystyle 1/x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/57e9787d3e7e96c76c9b0e247fecbf8d3b25335d height: height attribute not set width: width attribute not set description: {\displaystyle x\sim \gamma (\alpha ,\beta )\equiv \operatorname {gamma} (\alpha ,\beta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf760a328d5b468fea5f9f1d47cca54b558b6da height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ffad1626eab58fe191891edd57e7d27fa9a2def8 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma (\alpha )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/df21f16281b82ecb7a2dc5132fc54e88741b6855 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma (\alpha )=(\alpha -1)!} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6e29c7cfefa404f14fea26c52ea1471116570c height: height attribute not set width: width attribute not set description: {\displaystyle f(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\gamma (\alpha )}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0387febf90a45b820b486bf53b5a89a6d14fa664 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma (\alpha ,\beta x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c56150efa42bf73a899c72690ba1004ffcfa397e height: height attribute not set width: width attribute not set description: {\displaystyle f(x;\alpha ,\beta )=1-\sum _{i=0}^{\alpha -1}{\frac {(\beta x)^{i}}{i!}}e^{-\beta x}=e^{-\beta x}\sum _{i=\alpha }^{\infty }{\frac {(\beta x)^{i}}{i!}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b0320535d26cf0c3b96e123ab0a046b6db3b1f height: height attribute not set width: width attribute not set description: {\displaystyle x\sim \gamma (k,\theta )\equiv \operatorname {gamma} (k,\theta )} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/b/b1/gamma-pdf-3d.png/320px-gamma-pdf-3d.png height: 259 width: 320 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/caf176962d326ad7af8186d5f4cd3f3e7fae4852 height: height attribute not set width: width attribute not set description: {\displaystyle f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e10c41c86ac78205b3f7cd5a1ba84052714ff662 height: height attribute not set width: width attribute not set description: {\displaystyle f(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\gamma (k)}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7bab1607e067f732c5fc272613cb87e793c79af0 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b50954cd5d3328b10d9922b97b893c25cd76a8 height: height attribute not set width: width attribute not set description: {\displaystyle f(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}e^{-x/\theta }=e^{-x/\theta }\sum _{i=k}^{\infty }{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a58ddcce52712f203ad4142baba4a474b8bef8 height: height attribute not set width: width attribute not set description: {\displaystyle \mu =k\theta =\alpha /\beta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c88cc5e24c9d0eb7176ce48ec7069281aea69a2d height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/52db1d7831315dcadbff298e415c9fc48de55076 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7958ea261b36951e15f1c9363cd01a384a8217d9 height: height attribute not set width: width attribute not set description: {\displaystyle 2/{\sqrt {k}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/67b2480d1a1107cd132f94b5280db23ddde59033 height: height attribute not set width: width attribute not set description: {\displaystyle \mathrm {e} [x^{n}]=\theta ^{n}{\frac {\gamma (k+n)}{\gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/4/42/gamma_distribution_median_bounds.png/320px-gamma_distribution_median_bounds.png height: 240 width: 320 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/932c212790dbeb09a8b9244e3eeb9b8fbb31b309 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{\gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f287a8873fea8959af7d41cd891678d8875d3a15 height: height attribute not set width: width attribute not set description: {\displaystyle \theta =1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/50148b370f2224a88b71230d9b89d393ce2a35ae height: height attribute not set width: width attribute not set description: {\displaystyle k-{\frac {1}{3}}<\nu (k) |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a7dd26b6f7f7f39a301f14b3c05dff07882ba906 height: height attribute not set width: width attribute not set description: {\displaystyle \mu (k)=k} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7023bd901ef6b557d850ffa2089200e625d370e7 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0648b913658b37bf6c4db6034c038b8e385e2d height: height attribute not set width: width attribute not set description: {\displaystyle {\text{gamma}}(k,1)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e18aba75babe49bc0b1eb2a0e35726e395b93b height: height attribute not set width: width attribute not set description: {\displaystyle \mu =k\theta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af height: height attribute not set width: width attribute not set description: {\displaystyle \theta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/06425d870fa9b7a504eaaef775e51f7937ac4b24 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)=k-{\frac {1}{3}}+{\frac {8}{405k}}+{\frac {184}{25515k^{2}}}+{\frac {2248}{3444525k^{3}}}-{\frac {19006408}{15345358875k^{4}}}-o\left({\frac {1}{k^{5}}}\right)+\cdots } |
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https://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/gamma_distribution_median_lyon_bounds.png/320px-gamma_distribution_median_lyon_bounds.png height: 240 width: 320 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/de282e9589256ce4d8b25101b4a64b840fa4c820 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)\approx 2^{-1/k}(a+k)} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/gamma_distribution_median_loglog_bounds.png/320px-gamma_distribution_median_loglog_bounds.png height: 240 width: 320 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b height: height attribute not set width: width attribute not set description: {\displaystyle k=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8e0dbcff88172cb178dc51b01046ea6a4a0c9f height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45 height: height attribute not set width: width attribute not set description: {\displaystyle k>0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/89f7ba6d1b9eabdd7621fc99f96065e5b1f96dda height: height attribute not set width: width attribute not set description: {\displaystyle k2^{-1/k}<\nu (k) |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961 height: height attribute not set width: width attribute not set description: {\displaystyle k\geq 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/92b15d0fc8940dbe55f21e5697550d78722c4983 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)\leq k-1+\log 2~~} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa height: height attribute not set width: width attribute not set description: {\displaystyle k=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6aea83a57db153f846b8b7b1a8306e59c4e43ef0 height: height attribute not set width: width attribute not set description: {\displaystyle k=0.5} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4061f7fcbab5e475380fda6e2dfcf445d100e0dc height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b68a68b15e862c49b0f09a218ff1a94b061271e8 height: height attribute not set width: width attribute not set description: {\displaystyle k<1/9} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f59966fa882b4fdd8a8abbfe3882d7b0e5cbedda height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)\approx 2^{-1/k}(a+bk)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2591377ce5a7b571e43130a7aa0ae5e591c8bb height: height attribute not set width: width attribute not set description: {\displaystyle \nu _{l\infty }(k)=2^{-1/k}(\log 2-{\frac {1}{3}}+k)\quad } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5 height: height attribute not set width: width attribute not set description: {\displaystyle k\to \infty } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed520e224ba985e574fb77b71ee5bb6883512f2 height: height attribute not set width: width attribute not set description: {\displaystyle \nu _{u}(k)=2^{-1/k}(e^{-\gamma }+k)\quad } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcc764df099ea8bc091518756ccc1221f6e37ba height: height attribute not set width: width attribute not set description: {\displaystyle k\to 0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b201e900a30da19a1f1e4bdddcc70fe7e502be4b height: height attribute not set width: width attribute not set description: {\displaystyle e^{-x}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec07f2440f178b43a965d030b6d17daadbacfc2 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)>\left({\frac {2}{\gamma (k+1)}}\right)^{-1/k}\quad } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ad24afe34a23eb1560b491689eec1e7dea040c63 height: height attribute not set width: width attribute not set description: {\displaystyle \nu ^{\prime }(1)\approx 0.9680448} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4d3c22d938a07214c417f6975f7cbd17fbf1a9 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcb4e9ccbc2695aea8316f010e96d42d178e3d6 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)\geq \log 2+(k-1)(\gamma -2\operatorname {ei} (-\log 2)-\log \log 2)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f48b9898b0be0be38bbdfc4049164982bce09ea3 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (1)=\log 2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8435e11856bb84972296cd69082bd5ac49609938 height: height attribute not set width: width attribute not set description: {\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{l\infty }(k)+(1-{\tilde {g}}(k))\nu _{u}(k)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf709e979316ee494a3f076f7e1d97be44a3f8f height: height attribute not set width: width attribute not set description: {\displaystyle {\tilde {g}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77b991fb29325fbaa24480e075e54871f5128a62 height: height attribute not set width: width attribute not set description: {\displaystyle g(k)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ea545145faf7173d894259d15c1b2ed59cf70d height: height attribute not set width: width attribute not set description: {\displaystyle g(k)={\frac {\nu _{u}(k)-\nu (k)}{\nu _{u}(k)-\nu _{l\infty }(k)}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e65ab5d33035c86360b97f1dde6799ddeec9d4 height: height attribute not set width: width attribute not set description: {\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8e4d17f6dff2d8b4188f7d396887129affd9edb8 height: height attribute not set width: width attribute not set description: {\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8762f4ac41b74d0ac8c557125bda65e5431ee804 height: height attribute not set width: width attribute not set description: {\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/62d35fd0b7cd574ea4f7a49eb96f107f6ffe5edb height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{i=1}^{n}x_{i}\sim \mathrm {gamma} \left(\sum _{i=1}^{n}k_{i},\theta \right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/391514411e8eaf5fdc115ca83083262b1fc48424 height: height attribute not set width: width attribute not set description: {\displaystyle x\sim \mathrm {gamma} (k,\theta ),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/615789eea5ea246ac96af8627865abbad8f1b404 height: height attribute not set width: width attribute not set description: {\displaystyle cx\sim \mathrm {gamma} (k,c\,\theta ),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e3247407b9d18e6a48db04762641c9887835677c height: height attribute not set width: width attribute not set description: {\displaystyle x\sim \mathrm {gamma} \left(\alpha ,\beta \right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d2967d0b2f02976cb6572d8be0bb3281711d7e height: height attribute not set width: width attribute not set description: {\displaystyle cx\sim \mathrm {gamma} \left(\alpha ,{\frac {\beta }{c}}\right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/26e31141e3a3f0f1aa1ec728a7276269b702836c height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {e} [\ln x]=\psi (\alpha )-\ln \beta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ac33a1a15f14d72a259a487b16152938fd409770 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {e} [\ln x]=\psi (k)+\ln \theta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/35c82551037f1e1392e0d1dc34c8192aec01fe09 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {var} [\ln x]=\psi ^{(1)}(\alpha )=\psi ^{(1)}(k)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4f824d0b8ec54f716ece0f1d8acf4898cab2f7dd height: height attribute not set width: width attribute not set description: {\displaystyle \psi ^{(1)}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/61733af14d6071115cd2f558874cdac5180f6dfc height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\operatorname {h} (x)&=\operatorname {e} [-\ln p(x)]\\[4pt]&=\operatorname {e} [-\alpha \ln \beta +\ln \gamma (\alpha )-(\alpha -1)\ln x+\beta x]\\[4pt]&=\alpha -\ln \beta +\ln \gamma (\alpha )+(1-\alpha )\psi (\alpha ).\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c832644ca07117f067d7b85bff6540faf23e08 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {h} (x)=k+\ln \theta +\ln \gamma (k)+(1-k)\psi (k).} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/gamma-kl-3d.png/320px-gamma-kl-3d.png height: 209 width: 320 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c5567126eea1d9b2264fbc9b966d3621a739c5de height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}d_{\mathrm {kl} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \gamma (\alpha _{p})+\log \gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4bae9104706395e7b3d9da9f41c6b7a69b8d9423 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}d_{\mathrm {kl} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \gamma (k_{p})+\log \gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/81d5c9dac2cf6d0ae13d59b77058972caa492522 height: height attribute not set width: width attribute not set description: {\displaystyle f(s)=(1+\theta s)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d67872301909a9d739e265252ad0c7339cead069 height: height attribute not set width: width attribute not set description: {\displaystyle x_{1},x_{2},\ldots ,x_{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b height: height attribute not set width: width attribute not set description: {\displaystyle n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1973eff94821789d2a7ff09c2f5dc31be87f88d7 height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{i}x_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/81c7f94ec07eab6afc5d1110e916c1c73c980f13 height: height attribute not set width: width attribute not set description: {\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i}x_{i}\sim \operatorname {gamma} (n,n*\lambda )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cea8fc216dc5113f4d745dcee1fd769c4608dff6 height: height attribute not set width: width attribute not set description: {\displaystyle x\sim \gamma (k\in \mathbf {z} ,\theta ),\qquad y\sim \operatorname {pois} \left({\frac {x}{\theta }}\right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd3aee7ab286228c1c22b1ad3371d3f9aa2538a height: height attribute not set width: width attribute not set description: {\displaystyle p(x>x)=p(y |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/519b56baa947c79685e8946bb422b39654551779 height: height attribute not set width: width attribute not set description: {\displaystyle x^{2}\sim \gamma \left({\frac {3}{2}},2a^{2}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b112df3c2dc3c9c78d5b897a9d195737fd12371a height: height attribute not set width: width attribute not set description: {\textstyle \log x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/88a98f7181b035dbddcd947c5f1408b9c5ed23cd height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1d6f6db29b41fef2dcdc6f1cfee82644296c9e height: height attribute not set width: width attribute not set description: {\displaystyle a={\sqrt {\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5ff55e24541193805d60671f025c948707ce9933 height: height attribute not set width: width attribute not set description: {\displaystyle x^{q}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/482e0a33d9e8fd6307b5f68a5182c2d0d14efc9c height: height attribute not set width: width attribute not set description: {\displaystyle q>0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/142cecd27f4bc31247ef7624e1ab3c509f343039 height: height attribute not set width: width attribute not set description: {\displaystyle a=\theta ^{q}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f4bce9ca7bb5b08324fd5622ccee7d524aed9e height: height attribute not set width: width attribute not set description: {\displaystyle x_{k}\sim \gamma (\alpha _{k},\theta _{k})\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8dddf330ecfa29ac3839ed463ee66efe4e3f93 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\alpha _{2}\theta _{2}x_{1}}{\alpha _{1}\theta _{1}x_{2}}}\sim \mathrm {f} (2\alpha _{1},2\alpha _{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/18ad8b86fd514386d69012b06ed5a2724f3141eb height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {x_{1}}{x_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/93b3bb7f3563f635649d7dc74ad13f248e80bd53 height: height attribute not set width: width attribute not set description: {\displaystyle x_{k}\sim \gamma (\alpha _{k},\beta _{k})\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/89d4abcc11916af8cb166426737f02fc6feed39e height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\alpha _{2}\beta _{1}x_{1}}{\alpha _{1}\beta _{2}x_{2}}}\sim \mathrm {f} (2\alpha _{1},2\alpha _{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c8feeac0cc0fe85a8a361119b77b78cf628096f3 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {x_{1}}{x_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d857ffeb61fc56ae24b19d6a4dbe58e0b8b9f4c6 height: height attribute not set width: width attribute not set description: {\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\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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https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f318f1c6f5b6c50886d35fe09b9205c3e66784 height: height attribute not set width: width attribute not set description: {\displaystyle \psi (\alpha ,z)={}_{1}\psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0acf17a39a34862746426e99e25aa8ee005b3b height: height attribute not set width: width attribute not set description: {\displaystyle x|\theta \sim \gamma (k,\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7c9a793a31d07edd3f6d59d75202965b08b09d height: height attribute not set width: width attribute not set description: {\displaystyle \theta \sim ig(b,1)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d1cbed103b7527b8761b26fb2b729ad65844d7a6 height: height attribute not set width: width attribute not set description: {\displaystyle ig} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a350b021b5230bc13e56f87a39b1c64f816c04c3 height: height attribute not set width: width attribute not set description: {\displaystyle x\sim \beta '(k,b)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d14001f211d8e272b5c10e45c739d320359c48c8 height: height attribute not set width: width attribute not set description: {\displaystyle \beta '} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e5dddb7d9a9add51167f6c7786a39a798895f8 height: height attribute not set width: width attribute not set description: {\displaystyle f(x;k)\,(k>1)} |
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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/c7fdb1229d8546f9459b5bfdae27e98ee789a192 height: height attribute not set width: width attribute not set description: {\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,w_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {n}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/16503339cce78afe1b7b86dcf6d064fb7f34b979 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathfrak {n}}_{\alpha }(\nu )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe1c09916f6155dccb18cd323dcce7170ad5fcd height: height attribute not set width: width attribute not set description: {\displaystyle \alpha =1/k} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb78f76255ce2e2a500863818ce36691169fb9e height: height attribute not set width: width attribute not set description: {\displaystyle w_{k}(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/51e59f233200284ea82290f9fe3d1055c1f8bc02 height: height attribute not set width: width attribute not set description: {\displaystyle l(k,\theta )=\prod _{i=1}^{n}f(x_{i};k,\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d66eaf70de1c8f25f6531bc30e60177c104dba1e height: height attribute not set width: width attribute not set description: {\displaystyle \ell (k,\theta )=(k-1)\sum _{i=1}^{n}\ln x_{i}-\sum _{i=1}^{n}{\frac {x_{i}}{\theta }}-nk\ln \theta -n\ln \gamma (k)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b434bb6cc9b63502d363cd5396852735fb532cbf height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\theta }}={\frac {1}{kn}}\sum _{i=1}^{n}x_{i}={\frac {\bar {x}}{k}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2d3ca919f335d783d5ad3236b2411caa8d78c4 height: height attribute not set width: width attribute not set description: {\displaystyle \ell (k)=(k-1)\sum _{i=1}^{n}\ln x_{i}-nk-nk\ln \left({\frac {\sum x_{i}}{kn}}\right)-n\ln \gamma (k)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4a366b1cf2a65d58e6be08fede1a447e0771926b height: height attribute not set width: width attribute not set description: {\displaystyle n\geq 2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/85982022b9eb1f295b44de55023687a490db0a39 height: height attribute not set width: width attribute not set description: {\displaystyle n=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f050e906764d1b0f675a509a5a34a818f656aee0 height: height attribute not set width: width attribute not set description: {\displaystyle \ell (k)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/12fbe70b36b48cf309a3ddd33c2bda2288dc38c8 height: height attribute not set width: width attribute not set description: {\displaystyle \ln k-\psi (k)=\ln \left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}\right)-{\frac {1}{n}}\sum _{i=1}^{n}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/297c72b01e6457eae87c4b56f8bd06e8c4c3b61f height: height attribute not set width: width attribute not set description: {\displaystyle {\overline {\ln x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f8de4ca637e88113ae889e550bce8282c9654514 height: height attribute not set width: width attribute not set description: {\displaystyle \ln k-\psi (k)\approx {\frac {1}{2k}}\left(1+{\frac {1}{6k+1}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77d5e4e535c59fed0ec5f4f0c71c0abee169540f height: height attribute not set width: width attribute not set description: {\displaystyle s=\ln \left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}\right)-{\frac {1}{n}}\sum _{i=1}^{n}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eb19579e029ef52f2677c6aaa01d443a93dc4a85 height: height attribute not set width: width attribute not set description: {\displaystyle k\approx {\frac {3-s+{\sqrt {(s-3)^{2}+24s}}}{12s}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b13a1ddfbf19881de52a0f0e647b9d7c3dbf6c77 height: height attribute not set width: width attribute not set description: {\displaystyle k\leftarrow k-{\frac {\ln k-\psi (k)-s}{{\frac {1}{k}}-\psi ^{\prime }(k)}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7912fe7770f431c4e61a0e4595918a0848e49 height: height attribute not set width: width attribute not set description: {\displaystyle ({\hat {k}},{\hat {\theta }})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ed172b0f5195382a3500c713941f945ad4db3898 height: height attribute not set width: width attribute not set description: {\displaystyle \ln x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a77120c377a372035d5ef92b6e1be681791494fc height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln {\hat {\theta }}&={\overline {\ln x}}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe46c866379cec130e31f750cf08fa3ce5edfaa height: height attribute not set width: width attribute not set description: {\displaystyle (x_{1},\ldots ,x_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d52dc64c0a15754ca6e07f1772061da406e13308 height: height attribute not set width: width attribute not set description: {\displaystyle f(x;k,\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6de6fb21a1e6a5e2df503c4a69f91db17c876706 height: height attribute not set width: width attribute not set description: {\displaystyle p({\text{underflow}})=1-(1-f(\varepsilon ;k,\theta ))^{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ed55da0cb5f3e282ffc0d2c211ad2e7c2e24d978 height: height attribute not set width: width attribute not set description: {\displaystyle k=10^{-2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/92c893655aade068aa13c22966cb222d4bbac837 height: height attribute not set width: width attribute not set description: {\displaystyle n=10^{4}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/753624cf1c18b917bc51c09f9bc7261d1b1fbbe8 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon =2.25\times 10^{-308}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4145cd4e3821fab09005f66981c9745abd2ed752 height: height attribute not set width: width attribute not set description: {\displaystyle p({\text{underflow}})\approx 0.9998} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b098ac734780e67edb80b5a1039aea80f81595a1 height: height attribute not set width: width attribute not set description: {\displaystyle k<1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7e4951dd26ab192dfe5719aac257ae932942a1 height: height attribute not set width: width attribute not set description: {\displaystyle y\sim {\text{gamma}}(k+1,\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2d79a68d9de22a25a154cd29effa606ea01a52 height: height attribute not set width: width attribute not set description: {\displaystyle u\sim {\text{uniform}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8e3d978cd6908a457738820b19f2bb85e42929 height: height attribute not set width: width attribute not set description: {\displaystyle z=\ln(y)+\ln(u)/k} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ad42d845708982e924f81ef9a668a8c1464ba237 height: height attribute not set width: width attribute not set description: {\displaystyle \exp(z)\sim {\text{gamma}}(k,\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/76c91a0a26cb3f2fa4bc828a0c30cf81bd5fede8 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {k}}={\frac {n\sum _{i=1}^{n}x_{i}}{n\sum _{i=1}^{n}x_{i}\ln x_{i}-\sum _{i=1}^{n}x_{i}\sum _{i=1}^{n}\ln x_{i}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c6558c0ea90b9602485c6ba273feadb033fb4121 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\theta }}={\frac {1}{n^{2}}}\left(n\sum _{i=1}^{n}x_{i}\ln x_{i}-\sum _{i=1}^{n}x_{i}\sum _{i=1}^{n}\ln x_{i}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/65f9ada852433174f5c66beb07973e6809458a4c height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {k}}={\bar {x}}/{\hat {\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1993cfa1e3a5633d096f41763a1fb4a2dd588b3b height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\theta }}={\overline {x\ln {x}}}-{\bar {x}}{\overline {\ln {x}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/66836da08bc157ab0a5edf7974395e117eaee26c height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\beta }}=1/{\hat {\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/69e944316bfe95be86b91886fd1ed2c430835ccd height: height attribute not set width: width attribute not set description: {\displaystyle {\tilde {\theta }}={\frac {n}{n-1}}{\hat {\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a767a0e0a4372c8ae599487d3bcb2af62413fa8f height: height attribute not set width: width attribute not set description: {\displaystyle {\tilde {k}}={\hat {k}}-{\frac {1}{n}}\left(3{\hat {k}}-{\frac {2}{3}}\left({\frac {\hat {k}}{1+{\hat {k}}}}\right)-{\frac {4}{5}}{\frac {\hat {k}}{(1+{\hat {k}})^{2}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e88b49f3b39c1e9dfa4e3f11c9576df51513012d height: height attribute not set width: width attribute not set description: {\displaystyle p(\theta \mid k,x_{1},\dots ,x_{n})\propto {\frac {1}{\theta }}\prod _{i=1}^{n}f(x_{i};k,\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ffae7a36c6134471ab6b856eb7354a68f2ee8130 height: height attribute not set width: width attribute not set description: {\displaystyle y\equiv \sum _{i=1}^{n}x_{i},\qquad p(\theta \mid k,x_{1},\dots ,x_{n})=c(x_{i})\theta ^{-nk-1}e^{-y/\theta }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/99fbb69f9a62c02e04870f6859733934f0d6c087 height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{\infty }\theta ^{-nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{nk-1-m}e^{-xy}\,dx=y^{-(nk-m)}\gamma (nk-m)\!} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/61ae01ae77aa6c640cbaa1bb2a8863454827916a height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {e} [x^{m}]={\frac {\gamma (nk-m)}{\gamma (nk)}}y^{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/de03eb3d598f8c2c6481e9c44542904ee3858e9e height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {y}{nk-1}}\pm {\sqrt {\frac {y^{2}}{(nk-1)^{2}(nk-2)}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d513935bdc7db0e23f116aab5397368604572ccd height: height attribute not set width: width attribute not set description: {\displaystyle p(k,\theta \mid p,q,r,s)={\frac {1}{z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\gamma (k)^{r}\theta ^{ks}}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/404516eb571949b6c74db5cc102e9e5341a9de21 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p'&=p\prod \nolimits _{i}x_{i},\\q'&=q+\sum \nolimits _{i}x_{i},\\r'&=r+n,\\s'&=s+n,\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9961376aecd3518f5e3b88e2b02034dc36ab8e4f height: height attribute not set width: width attribute not set description: {\displaystyle \alpha =n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/88a83a9843f29cf470dc2831bda24053d4262bb9 height: height attribute not set width: width attribute not set description: {\displaystyle -\sum _{k=1}^{n}\ln u_{k}\sim \gamma (n,1)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/78687aeaefa95f26e5c2eedf29a5e9d16e354292 height: height attribute not set width: width attribute not set description: {\displaystyle u\leq {\frac {e}{e+\delta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fc600e64975b15e012887f0072dd610e9ee928ab height: height attribute not set width: width attribute not set description: {\displaystyle \xi =v^{1/\delta }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9acf3fb974231242cb1622667fb3fef4dbd65b height: height attribute not set width: width attribute not set description: {\displaystyle \eta =w\xi ^{\delta -1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/01c3bb894921726acc9b91410489f8db898cddd5 height: height attribute not set width: width attribute not set description: {\displaystyle \xi =1-\ln v} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b92760858b460290e1acc45654f8713b8770ed height: height attribute not set width: width attribute not set description: {\displaystyle \eta =we^{-\xi }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c999d5c04f901d0d448bed573cee6b42166dd46f height: height attribute not set width: width attribute not set description: {\displaystyle \eta >\xi ^{\delta -1}e^{-\xi }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a6adf8db6aaf40d7b1cd9fa13983dac96ce6f2fa height: height attribute not set width: width attribute not set description: {\displaystyle \theta \left(\xi -\sum _{i=1}^{\lfloor k\rfloor }\ln u_{i}\right)\sim \gamma (k,\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/51f6297fc6736d117d930b473bc9d4325f8228ad height: height attribute not set width: width attribute not set description: {\displaystyle \scriptstyle \lfloor k\rfloor } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b5f13d36077c9400af25d998bbe4bffa7772c5 height: height attribute not set width: width attribute not set description: {\displaystyle d=a-{\frac {1}{3}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e74964f3d4c7c5a61d85bd18e36b9c6bedf0509d height: height attribute not set width: width attribute not set description: {\displaystyle c={\frac {1}{\sqrt {9d}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd133c8bb01c03e81f752a57d132ee59fcff2ea height: height attribute not set width: width attribute not set description: {\displaystyle v=(1+cx)^{3}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c314fc908a83c555d34968d25e86c5ae0b76ef6f height: height attribute not set width: width attribute not set description: {\displaystyle v>0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ae59d8c5427259a4388fd99891925d4d8db16821 height: height attribute not set width: width attribute not set description: {\displaystyle \ln u<{\frac {x^{2}}{2}}+d-dv+d\ln v} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5f695cb4931398dd078f0335e6de6905abf748 height: height attribute not set width: width attribute not set description: {\displaystyle dv} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8d12ad014d8646f91b7db9d53dd0a4bf1afdab height: height attribute not set width: width attribute not set description: {\displaystyle 1\leq a=\alpha =k} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a438686f43515309dc233e2bb3b2c8344357c7ef height: height attribute not set width: width attribute not set description: {\displaystyle \gamma _{\alpha }=\gamma _{1+\alpha }u^{1/\alpha }} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/wikibooks-logo-en-noslogan.svg/40px-wikibooks-logo-en-noslogan.svg.png height: 40 width: 40 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/en/thumb/9/96/symbol_category_class.svg/16px-symbol_category_class.svg.png height: 16 width: 16 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/commons-logo.svg/12px-commons-logo.svg.png height: 16 width: 12 description: no alt description found |
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https://login.wikimedia.org/wiki/special:centralautologin/start?type=1x1 height: 1 width: 1 description: no alt description found |
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http://en.wikipedia.org/static/images/footer/wikimedia-button.svg height: 29 width: 84 description: wikimedia foundation |
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http://en.wikipedia.org/w/resources/assets/poweredby_mediawiki.svg height: 31 width: 88 description: powered by mediawiki |
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