Probability density function
Cumulative distribution function
|Parameters||(degrees of freedom)|
|MGF||Complicated (see text)|
|CF||Complicated (see text)|
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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 ).
Probability density functionEdit
The probability density function is
where is the Gamma function.
Cumulative distribution functionEdit
The cumulative distribution function is given by:
where is the regularized Gamma function.
Moment generating functionEdit
The moment generating function is given by:
where is Kummer's confluent hypergeometric function.
The characteristic function is given by:
where again, is Kummer's confluent hypergeometric function.
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:
The entropy is given by:
where is the polygamma function.
- If then (chi-squared distribution)
- (Normal distribution)
- If then
- If then (half-normal distribution) for any
- (Rayleigh distribution)
- (Maxwell distribution)
- (The 2-norm of standard normally distributed variables is a chi distribution with degrees of freedom)
- chi distribution is a special case of the generalized gamma distribution or the Nakagami distribution or the noncentral chi distribution
|noncentral chi-squared distribution|
|noncentral chi distribution|