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| Probability density function|
| Cumulative distribution function|
|Parameters||(degrees of freedom)|
|mgf||Complicated (see text)|
|Char. func.||Complicated (see text)|
In probability theory and statistics, the chi distribution is a continuous probability distribution. The distribution usually arises when a k-dimensional vector's orthogonal components are independent and each follow a standard normal distribution. The length of the vector will then have a chi distribution. The most familiar example is the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom. If are k independent, normally distributed random variables with means and standard deviations , then the statistic
is distributed according to the chi distribution. The chi distribution has one parameter: which specifies the number of degrees of freedom (i.e. the number of ).
The probability density function is
where is the Gamma function. The cumulative distribution function is 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 characteristic function is given by:
where again, is Kummer's confluent hypergeometric function. The entropy is given by:
where is the polygamma function.
- If is chi distributed then is chi-square distributed:
- The Rayleigh distribution with is a chi distribution with two degrees of freedom.
- The Maxwell distribution for normalized molecular speeds is a chi distribution with three degrees of freedom.
- The chi distribution for is the half-normal distribution.
|noncentral chi-square distribution|
|noncentral chi distribution|
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