{\Gamma(k/2)}| cdf =$P(k/2,x^2/2)\backslash ,$| mean =$\backslash mu=\backslash sqrt\{2\}\backslash ,\backslash frac\{\backslash Gamma((k+1)/2)\}\{\backslash Gamma(k/2)\}$| median =| mode =$\backslash sqrt\{k-1\}\backslash ,$ for $k\backslash ge\; 1$| variance =$\backslash sigma^2=k-\backslash mu^2\backslash ,$| skewness =$\backslash gamma\_1=\backslash frac\{\backslash mu\}\{\backslash sigma^3\}\backslash ,(1-2\backslash sigma^2)$| kurtosis =$\backslash frac\{2\}\{\backslash sigma^2\}(1-\backslash mu\backslash sigma\backslash gamma\_1-\backslash sigma^2)$| entropy =$\backslash ln(\backslash Gamma(k/2))+\backslash ,$
$\backslash frac\{1\}\{2\}(k\backslash !-\backslash !\backslash ln(2)\backslash !-\backslash !(k\backslash !-\backslash !1)\backslash psi\_0(k/2))$| mgf =Complicated (see text)| char =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 $\backslash mu\_i$ and variances $\backslash sigma\_i$, then the statistic

$Z\; =\; \backslash sqrt\{\backslash sum\_1^k\; \backslash left(\backslash frac\{X\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^2\}$

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 $X\_i$).

Properties

---

The probability density function is

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

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

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

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

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

$t\backslash sqrt\{2\}\backslash ,\backslash frac\{\backslash Gamma((k+1)/2)\}\{\backslash Gamma(k/2)\}$

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:

$\backslash mu\_j=2^\{j/2\}\backslash frac\{\backslash Gamma((k+j)/2)\}\{\backslash Gamma(k/2)\}$

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

$\backslash mu\_1=\backslash sqrt\{2\}\backslash ,\backslash ,\backslash frac\{\backslash Gamma((k\backslash !+\backslash !1)/2)\}\{\backslash Gamma(k/2)\}$

$\backslash mu\_2=k\backslash ,$

$\backslash mu\_3=2\backslash sqrt\{2\}\backslash ,\backslash ,\backslash frac\{\backslash Gamma((k\backslash !+\backslash !3)/2)\}\{\backslash Gamma(k/2)\}=(k+1)\backslash mu\_1$

$\backslash mu\_4=(k)(k+2)\backslash ,$

$\backslash mu\_5=4\backslash sqrt\{2\}\backslash ,\backslash ,\backslash frac\{\backslash Gamma((k\backslash !+\backslash !5)/2)\}\{\backslash Gamma(k/2)\}=(k+1)(k+3)\backslash mu\_1$

$\backslash mu\_6=(k)(k+2)(k+4)\backslash ,$

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

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

From these expressions we may derive the following relationships:

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

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

Skewness: $\backslash gamma\_1=\backslash frac\{\backslash mu\}\{\backslash sigma^3\}\backslash ,(1-2\backslash sigma^2)$

Kurtosis excess: $\backslash gamma\_2=\backslash frac\{2\}\{\backslash sigma^2\}(1-\backslash mu\backslash sigma\backslash gamma\_1-\backslash sigma^2)$

The characteristic function is given by:

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

$it\backslash sqrt\{2\}\backslash ,\backslash frac\{\backslash Gamma((k+1)/2)\}\{\backslash Gamma(k/2)\}$

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:

$S=\backslash ln(\backslash Gamma(k/2))+\backslash frac\{1\}\{2\}(k\backslash !-\backslash !\backslash ln(2)\backslash !-\backslash !(k\backslash !-\backslash !1)\backslash psi\_0(k/2))$

where $\backslash psi\_0(z)$ is the Polygamma function.

Related distributions

---

• If $X$ is chi distributed $X\; \backslash sim\; \backslash chi\_k(x)$ then $X^2$ is chi-square distributed: $X^2\; \backslash sim\; \backslash chi^2\_k$
• The Rayleigh distribution with $\backslash sigma=1$ 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	$\backslash sum\_1^k\; \backslash left(\backslash frac\{X\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^2$
noncentral chi-square distribution	$\backslash sum\_1^k\; \backslash left(\backslash frac\{X\_i\}\{\backslash sigma\_i\}\backslash right)^2$
chi distribution	$\backslash sqrt\{\backslash sum\_1^k\; \backslash left(\backslash frac\{X\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^2\}$
noncentral chi distribution	$\backslash sqrt\{\backslash sum\_1^k\; \backslash left(\backslash frac\{X\_i\}\{\backslash sigma\_i\}\backslash right)^2\}$

Continuous distributions