Chi distribution


In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from 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 are independent, normally distributed random variables with mean 0 and standard deviation 1, then the statistic
is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution. The chi distribution has one parameter,, which specifies the number of degrees of freedom.
The most familiar examples are the Rayleigh distribution and the Maxwell–Boltzmann distribution of the molecular speeds in an ideal gas.

Definitions

Probability density function

The probability density function of the chi-distribution is
where is the gamma function.

Cumulative distribution function

The cumulative distribution function is given by:
where is the regularized gamma function.

Generating functions

The moment-generating function is given by:
where is Kummer's confluent hypergeometric function. The characteristic function is given by:

Properties

Moments

The raw moments are then given by:
where is the gamma function. Thus the first few raw moments are:
where the rightmost expressions are derived using the recurrence relationship for the gamma function:
From these expressions we may derive the following relationships:
Mean:
Variance:
Skewness:
Kurtosis excess:

Entropy

The entropy is given by:
where is the polygamma function.

Large n approximation

We find the large n=k+1 approximation of the mean and variance of chi distribution. This has application e.g. in finding the distribution of standard deviation of a sample of normally distributed population, where n is the sample size.
The mean is then:
We use the following Gamma function equality:
To write:
Using Stirling's approximation for Gamma function, we get the following expression for the mean:
And thus the variance is:

Related distributions


NameStatistic
chi-squared distribution
noncentral chi-squared distribution
chi distribution
noncentral chi distribution