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\ge 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. 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

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

External links