Chi-square distribution

chi-square
	Probability density function; <img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Chi-square_distributionPDF.png/300px-Chi-square_distributionPDF.png" decoding="async" width="300" height="225" class="mw-file-element" data-file-width="1300" data-file-height="975">
	Cumulative distribution function; <img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Chi-square_distributionCDF.png/300px-Chi-square_distributionCDF.png" decoding="async" width="300" height="225" class="mw-file-element" data-file-width="1300" data-file-height="975">
Parameters	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0b7004b5d82ad9beec2ad8de5cd31f33be4944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.859ex; height:2.176ex;" alt="{\displaystyle k>0\,}"> degrees of freedom
Support	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e777dea7f43a5ce0450f3a3644f27ea5e0b8cb90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.437ex; height:2.843ex;" alt="{\displaystyle x\in [0;+\infty )\,}">
Probability density function (pdf)	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c9eaa04356417ba1c2cf586a51b9612c1cb204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.594ex; height:6.843ex;" alt="{\displaystyle {\frac {(1/2)^{k/2}}{\Gamma (k/2)}}x^{k/2-1}e^{-x/2}\,}">
Cumulative distribution function (cdf)	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450090ab05cc3987d7181080c2cc0252efb37f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.52ex; height:6.509ex;" alt="{\displaystyle {\frac {\gamma (k/2,x/2)}{\Gamma (k/2)}}\,}">
Mean	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5665cb00a844c1c6a671c119c6e4984d28fb851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.598ex; height:2.176ex;" alt="{\displaystyle k\,}">
Median	approximately <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794742e6d045dd68392fbbe147b54c4059899f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle k-2/3\,}">
Mode	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9117481e8c57d332342de9f8006b9119196c56f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.601ex; height:2.343ex;" alt="{\displaystyle k-2\,}"> if <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4b1e426322c6f030c7fb0e1fbbede6215febef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.859ex; height:2.343ex;" alt="{\displaystyle k\geq 2\,}">
Variance	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf315d85ad773a5a895e94e8221f6f29ce38d4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.148ex; height:2.176ex;" alt="{\displaystyle 2\,k\,}">
Skewness	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2194d3fcf2f7d668d726161d8d28b6892da42b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.247ex; height:4.843ex;" alt="{\displaystyle {\sqrt {8/k}}\,}">
Excess kurtosis	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dfb2478b228cc67f13c4b63cc94bf4fd0ca2e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.086ex; height:2.843ex;" alt="{\displaystyle 12/k\,}">
Entropy	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aec8310865abcb4726d4edfc7a9640bc20cd50f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.322ex; height:5.343ex;" alt="{\displaystyle {\frac {k}{2}}\!+\!\ln(2\Gamma (k/2))\!+\!(1\!-\!k/2)\psi (k/2)}">
Moment-generating function (mgf)	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb18c03d0811ce0ff4fcde251fe3ecffb90afa95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.213ex; height:3.343ex;" alt="{\displaystyle (1-2\,t)^{-k/2}}"> for <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b0bb14998cb35dfc1287f85405b08e87091947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.037ex; height:2.176ex;" alt="{\displaystyle 2\,t<1\,}">
Characteristic function	<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d153a5cbb8c72c28bafd6ef41135944f41013ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.789ex; height:3.343ex;" alt="{\displaystyle (1-2\,i\,t)^{-k/2}\,}">

In probability theory and statistics, the chi-square distribution (also chi-squared or $\chi ^{2}$ distribution) is one of the most widely used theoretical probability distributions. Chi-square distribution with $\nu$ degrees of freedom is written as $\chi ^{2}(\nu )$ .^[1] It is a special case of gamma distribution.^[2]

Chi-square distribution is primarily used in statistical significance tests and confidence intervals.^[3] It is useful, because it is relatively easy to show that certain probability distributions come close to it, under certain conditions. One of these conditions is that the null hypothesis must be true. Another one is that the different random variables (or observations) must be independent of each other.

Related pages

Chi-squared test

References

Other websites

Chi-Square Tutorial by Khan Academy

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[1]

[2]

[3]

Probability density function
Cumulative distribution function
Parameters	$k>0\,$ degrees of freedom
Support	$x\in [0;+\infty )\,$
Probability density function (pdf)	${\frac {(1/2)^{k/2}}{\Gamma (k/2)}}x^{k/2-1}e^{-x/2}\,$
Cumulative distribution function (cdf)	${\frac {\gamma (k/2,x/2)}{\Gamma (k/2)}}\,$
Mean	$k\,$
Median	approximately $k-2/3\,$
Mode	$k-2\,$ if $k\geq 2\,$
Variance	$2\,k\,$
Skewness	${\sqrt {8/k}}\,$
Excess kurtosis	$12/k\,$
Entropy	${\frac {k}{2}}\!+\!\ln(2\Gamma (k/2))\!+\!(1\!-\!k/2)\psi (k/2)$
Moment-generating function (mgf)	$(1-2\,t)^{-k/2}$ for $2\,t<1\,$
Characteristic function	$(1-2\,i\,t)^{-k/2}\,$