Chi distribution information

chi
	Probability density function;
	Cumulative distribution function;
Notation	or
Parameters	(degrees of freedom)
Support
PDF
CDF
Mean
Median
Mode	for
Variance
Skewness
Excess kurtosis
Entropy	;
MGF	Complicated (see text)
CF	Complicated (see text)

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. Equivalently, it is the distribution of the Euclidean distance between a multivariate Gaussian random variable and the origin. It is thus related to the chi-squared distribution by describing the distribution of the positive square roots of a variable obeying a chi-squared distribution.

If $Z_{1},\ldots ,Z_{k}$ are $k$ independent, normally distributed random variables with mean 0 and standard deviation 1, then the statistic

Y={\sqrt {\sum _{i=1}^{k}Z_{i}^{2}}}

is distributed according to the chi distribution. The chi distribution has one positive integer parameter $k$ , which specifies the degrees of freedom (i.e. the number of random variables $Z_{i}$ ).

The most familiar examples are the Rayleigh distribution (chi distribution with two degrees of freedom) and the Maxwell–Boltzmann distribution of the molecular speeds in an ideal gas (chi distribution with three degrees of freedom).

Chi distribution information

and 16 Related for: Chi distribution information

Chi distribution

Noncentral chi distribution

Rayleigh distribution

List of probability distributions

Chi

Normal distribution

Gamma distribution

Nakagami distribution

Noncentral distribution

Multivariate normal distribution

Exponential distribution

Standard deviation

Poisson distribution

Wishart distribution

Ratio distribution

List of statistics articles

Probability density function
Cumulative distribution function
Notation	$\chi (k)\;$ or $\chi _{k}\!$
Parameters	$k>0\,$ (degrees of freedom)
Support	$x\in [0,\infty )$
PDF	${\frac {1}{2^{(k/2)-1}\Gamma (k/2)}}\;x^{k-1}e^{-x^{2}/2}$
CDF	$P(k/2,x^{2}/2)\,$
Mean	$\mu ={\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}$
Median	$\approx {\sqrt {k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}}}$
Mode	${\sqrt {k-1}}\,$ for $k\geq 1$
Variance	$\sigma ^{2}=k-\mu ^{2}\,$
Skewness	$\gamma _{1}={\frac {\mu }{\sigma ^{3}}}\,(1-2\sigma ^{2})$
Excess kurtosis	${\frac {2}{\sigma ^{2}}}(1-\mu \sigma \gamma _{1}-\sigma ^{2})$
Entropy	$\ln(\Gamma (k/2))+\,$ ${\frac {1}{2}}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi _{0}(k/2))$
MGF	Complicated (see text)
CF	Complicated (see text)