Chi distribution

For a broader coverage related to this topic, see Chi-squared 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	${\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}$

External links

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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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