# Chi distribution

Parameters Probability density function Cumulative distribution function $k>0\,$ (degrees of freedom) $x\in [0;\infty)$ $\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}$ $P(k/2,x^2/2)\,$ $\mu=\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}$ $\sqrt{k-1}\,$ for $k\ge 1$ $\sigma^2=k-\mu^2\,$ $\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)$ $\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)$ $\ln(\Gamma(k/2))+\,$ $\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))$ Complicated (see text) Complicated (see text)

In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the square root of the sum of squares of independent random variables having a standard normal distribution. The most familiar example is the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom (one for each spatial coordinate). If $X_i$ are k independent, normally distributed random variables with means $\mu_i$ and standard deviations $\sigma_i$, then the statistic

$Y = \sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$

is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution (scaled by the square root of n − 1) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution. The chi distribution has one parameter: $k$ which specifies the number of degrees of freedom (i.e. the number of $X_i$).

## Characterization

### Probability density function

The probability density function is

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

where $\Gamma(z)$ is the Gamma function.

### Cumulative distribution 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.

### Generating functions

#### Moment generating function

The moment generating function is given by:

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

#### Characteristic function

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.

## Properties

Differential equation

$\left\{x f'(x)+f(x) \left(-\nu +x^2+1\right)=0,f(1)=\frac{2^{1-\frac{\nu }{2}}}{\sqrt{e} \Gamma \left(\frac{\nu }{2}\right)}\right\}$

### Moments

The raw moments are then given by:

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

where $\Gamma(z)$ 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: $\gamma_2=\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)$

### Entropy

The entropy is given by:

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

where $\psi_0(z)$ is the polygamma function.

## Related distributions

• If $X \sim \chi_k(x)$ then $X^2 \sim \chi^2_k$ (chi-squared distribution)
• $\lim_{k \to \infty}\tfrac{\chi_k(x)-\mu_k}{\sigma_k} \xrightarrow{d}\ N(0,1) \,$ (Normal distribution)
• If $X \sim N(0,1)\,$ then $| X | \sim \chi_1(x) \,$
• If $X \sim \chi_1(x) \,$ then $\sigma X \sim HN(\sigma)\,$ (half-normal distribution) for any $\sigma > 0 \,$
• $\chi_2(x) \sim \mathrm{Rayleigh}(1)\,$ (Rayleigh distribution)
• $\chi_3(x) \sim \mathrm{Maxwell}(1)\,$ (Maxwell distribution)
• $\|\boldsymbol{N}_{i=1,\ldots,k}{(0,1)}\|_2 \sim \chi_k(x)$ (The 2-norm of $k$ standard normally distributed variables is a chi distribution with $k$ degrees of freedom)
• chi distribution is a special case of the generalized gamma distribution or the nakagami distribution or the noncentral chi distribution
Various chi and chi-squared distributions
Name Statistic
chi-squared distribution $\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-squared distribution $\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution $\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution $\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}$