# Chi-square distribution

continuous probability distribution

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

## References

1. "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-09-14.
2. Weisstein, Eric W. "Chi-Squared Distribution". mathworld.wolfram.com. Retrieved 2020-09-14.
3. "1.3.6.6.6. Chi-Square Distribution". www.itl.nist.gov. Retrieved 2020-09-14.
Parameters Probability density function Cumulative distribution function ${\displaystyle k>0\,}$  degrees of freedom ${\displaystyle x\in [0;+\infty )\,}$ ${\displaystyle {\frac {(1/2)^{k/2}}{\Gamma (k/2)}}x^{k/2-1}e^{-x/2}\,}$ ${\displaystyle {\frac {\gamma (k/2,x/2)}{\Gamma (k/2)}}\,}$ ${\displaystyle k\,}$ approximately ${\displaystyle k-2/3\,}$ ${\displaystyle k-2\,}$  if ${\displaystyle k\geq 2\,}$ ${\displaystyle 2\,k\,}$ ${\displaystyle {\sqrt {8/k}}\,}$ ${\displaystyle 12/k\,}$ ${\displaystyle {\frac {k}{2}}\!+\!\ln(2\Gamma (k/2))\!+\!(1\!-\!k/2)\psi (k/2)}$ ${\displaystyle (1-2\,t)^{-k/2}}$  for ${\displaystyle 2\,t<1\,}$ ${\displaystyle (1-2\,i\,t)^{-k/2}\,}$