# Chi distribution

For a broader coverage related to this topic, see Chi-squared distribution.
Parameters Probability density function Cumulative distribution function (degrees of freedom) for Complicated (see text) 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 are k independent, normally distributed random variables with means and standard deviations , then the statistic

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: which specifies the number of degrees of freedom (i.e. the number of ).

## Characterization

### Probability density function

The probability density function is

where is the Gamma function.

### Cumulative distribution function

The cumulative distribution function is given by:

where is the regularized Gamma function.

### Generating functions

#### Moment generating function

The moment generating function is given by:

#### Characteristic function

The characteristic function is given by:

where again, is Kummer's confluent hypergeometric function.

## Properties

Differential equation

### Moments

The raw moments are then given by:

where is the Gamma function. 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.

## Related distributions

Various chi and chi-squared distributions
Name Statistic
chi-squared distribution
noncentral chi-squared distribution
chi distribution
noncentral chi distribution