Chi distribution

chi

Probability density function
Cumulative distribution function
Parameters	$k>0\,$ (degrees of freedom)
Support	$x\in [0;\infty )$
PDF	${\frac {2^{{1-k/2}}x^{{k-1}}e^{{-x^{2}/2}}}{\Gamma (k/2)}}$
CDF	$P(k/2,x^{2}/2)\,$
Mean	$\mu ={\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}$
Mode	${\sqrt {k-1}}\,$ for $k\geq 1$
Variance	$\sigma ^{2}=k-\mu ^{2}\,$
Skewness	$\gamma _{1}={\frac {\mu }{\sigma ^{3}}}\,(1-2\sigma ^{2})$
Ex. kurtosis	${\frac {2}{\sigma ^{2}}}(1-\mu \sigma \gamma _{1}-\sigma ^{2})$
Entropy	$\ln(\Gamma (k/2))+\,$ ${\frac {1}{2}}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi _{0}(k/2))$
MGF	Complicated (see text)
CF	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, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. The most familiar examples are the Rayleigh distribution with chi distribution with 2 degrees of freedom, and 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

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

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 $X^{2}\sim \chi _{k}^{2}$ (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	${\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}$

See also

• Nakagami distribution

External links

• http://mathworld.wolfram.com/ChiDistribution.html