Chi distribution

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chi
Probability density function
Cumulative distribution function
Parameters	$k>0\,$ (degrees of freedom)
Support	$x\in [0;\infty)$
Probability density function (pdf)	$\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}$
Cumulative distribution function (cdf)	$P(k/2,x^2/2)\,$
Mean	$\mu=\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}$
Median
Mode	$\sqrt{k-1}\,$ for $k\ge 1$
Variance	$\sigma^2=k-\mu^2\,$
Skewness	$\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)$
Excess kurtosis	$\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)$
Entropy	$\ln(\Gamma(k/2))+\,$ $\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))$
Moment-generating function (mgf)	Complicated (see text)
Characteristic function	Complicated (see text)

In probability theory and statistics, the chi distribution is a continuous probability distribution. The distribution usually arises when a k-dimensional vector's orthogonal components are independent and each follow a standard normal distribution. The length of the vector will then have a chi distribution. The most familiar example is the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom. If $X i$ are k independent, normally distributed random variables with means $μ i$ and standard deviations $σ i$ , then the statistic

$Z = \sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$

is distributed according to the chi distribution. The chi distribution has one parameter: $k$ which specifies the number of degrees of freedom (i.e. the number of $X i$ ).

[edit] Properties

The probability density function is

$f(x;k) = \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}$

where $Γ(z)$ is the Gamma function. The cumulative distribution function is given by:

$F(x;k)=P(k/2,x^2/2)\,$

where $P (k, x)$ is the regularized Gamma function. The moment generating function is given by:

$M(t)=M\left(\frac{k}{2},\frac{1}{2},\frac{t^2}{2}\right)+$

$t\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)} M\left(\frac{k+1}{2},\frac{3}{2},\frac{t^2}{2}\right)$

where $M (a, b, z)$ is Kummer's confluent hypergeometric function. The raw moments are then given by:

$\mu_j=2^{j/2}\frac{\Gamma((k+j)/2)}{\Gamma(k/2)}$

where $Γ(z)$ is the Gamma function. The first few raw moments are:

$\mu_1=\sqrt{2}\,\,\frac{\Gamma((k\!+\!1)/2)}{\Gamma(k/2)}$

$\mu_2=k\,$

$\mu_3=2\sqrt{2}\,\,\frac{\Gamma((k\!+\!3)/2)}{\Gamma(k/2)}=(k+1)\mu_1$

$\mu_4=(k)(k+2)\,$

$\mu_5=4\sqrt{2}\,\,\frac{\Gamma((k\!+\!5)/2)}{\Gamma(k/2)}=(k+1)(k+3)\mu_1$

$\mu_6=(k)(k+2)(k+4)\,$

where the rightmost expressions are derived using the recurrence relationship for the Gamma function:

$\Gamma(x+1)=x\Gamma(x)\,$

From these expressions we may derive the following relationships:

Mean: $\mu=\sqrt{2}\,\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}$

Variance: $\sigma^2=k-\mu^2\,$

Skewness: $\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)$

Kurtosis excess: $\gamma_2=\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)$

The characteristic function is given by:

$\varphi(t;k)=M\left(\frac{k}{2},\frac{1}{2},\frac{-t^2}{2}\right)+$

$it\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)} M\left(\frac{k+1}{2},\frac{3}{2},\frac{-t^2}{2}\right)$

where again, $M (a, b, z)$ is Kummer's confluent hypergeometric function. The entropy is given by:

$S=\ln(\Gamma(k/2))+\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))$

where $ψ 0 (z)$ is the polygamma function.

[edit] Related distributions

If $X$ is chi distributed $X \sim \chi_k(x)$ then $X 2$ is chi-square distributed: $X^2 \sim \chi^2_k$
The Rayleigh distribution with $σ = 1$ is a chi distribution with two degrees of freedom.
The Maxwell distribution for normalized molecular speeds is a chi distribution with three degrees of freedom.
The chi distribution for $k = 1$ is the half-normal distribution.

**Various chi and chi-square distributions**
Name	Statistic
chi-square distribution	$\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-square distribution	$\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution	$\sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution	$\sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}$