Chi-square distribution

chi-square
	Probability density function;
	Cumulative distribution function;
Parameters	k>0 degrees of freedom
Support	x∈[0;+∞)
Probability density function (pdf)	(1/2)k/2Γ(k/2)xk/2−1e−x/2
Cumulative distribution function (cdf)	γ(k/2,x/2)Γ(k/2)
Mean	k
Median	approximately k−2/3
Mode	k−2 if k≥2
Variance	2k
Skewness	8/k
Excess kurtosis	12/k
Entropy	k2+ln⁡(2Γ(k/2))+(1−k/2)ψ(k/2)
Moment-generating function (mgf)	(1−2t)−k/2 for 2t<1
Characteristic function	(1−2it)−k/2

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

Chi-square distribution is primarily used in statistical significance tests and confidence intervals. 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

↑ "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.

Other websites

Chi-Square Tutorial by Khan Academy

This short article about mathematics can be made longer. You can help Wikipedia by adding to it.

[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.

Probability density function
Cumulative distribution function
Parameters	$k > 0$ degrees of freedom
Support	$x \in [0; + \infty)$
Probability density function (pdf)	$\frac{(1 / 2)^{k / 2}}{Γ (k / 2)} x^{k / 2 - 1} e^{- x / 2}$
Cumulative distribution function (cdf)	$\frac{γ (k / 2, x / 2)}{Γ (k / 2)}$
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	$\frac{k}{2} + \ln (2 Γ (k / 2)) + (1 - k / 2) ψ (k / 2)$
Moment-generating function (mgf)	$(1 - 2 t)^{- k / 2}$ for $2 t < 1$
Characteristic function	$(1 - 2 i t)^{- k / 2}$