Chi-square distribution
In probability theory and statistics, the chi-square distribution (also chi-squared or [math]\displaystyle{ \chi^2 }[/math] distribution) is one of the most widely used theoretical probability distributions. Chi-square distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom is written as [math]\displaystyle{ \chi^2(\nu) }[/math].[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.
Chi-square Distribution Media
Chernoff bound for the CDF and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (k = 10)
Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference (blue) and relative difference (red) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.
Related pages
References
- ↑ "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-09-14.
- ↑ Weisstein, Eric W. "Chi-Squared Distribution". mathworld.wolfram.com. Retrieved 2020-09-14.
- ↑ "1.3.6.6.6. Chi-Square Distribution". www.itl.nist.gov. Retrieved 2020-09-14.
Probability density function | |
Cumulative distribution function | |
Parameters | [math]\displaystyle{ k \gt 0\, }[/math] degrees of freedom |
---|---|
Support | [math]\displaystyle{ x \in [0; +\infty)\, }[/math] |
Probability density function (pdf) | [math]\displaystyle{ \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\, }[/math] |
Cumulative distribution function (cdf) | [math]\displaystyle{ \frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\, }[/math] |
Mean | [math]\displaystyle{ k\, }[/math] |
Median | approximately [math]\displaystyle{ k-2/3\, }[/math] |
Mode | [math]\displaystyle{ k-2\, }[/math] if [math]\displaystyle{ k\geq 2\, }[/math] |
Variance | [math]\displaystyle{ 2\,k\, }[/math] |
Skewness | [math]\displaystyle{ \sqrt{8/k}\, }[/math] |
Excess kurtosis | [math]\displaystyle{ 12/k\, }[/math] |
Entropy | [math]\displaystyle{ \frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2) }[/math] |
Moment-generating function (mgf) | [math]\displaystyle{ (1-2\,t)^{-k/2} }[/math] for [math]\displaystyle{ 2\,t\lt 1\, }[/math] |
Characteristic function | [math]\displaystyle{ (1-2\,i\,t)^{-k/2}\, }[/math] |