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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

Probability density function
Plot of the Rayleigh PMF
Cumulative distribution function
Plot of the Rayleigh CMF
Parameters (degrees of freedom)
Mode for
Ex. kurtosis
MGF Complicated (see text)
CF Complicated (see text)

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.

Generating functionsEdit

Moment generating functionEdit

The moment generating function is given by:


where   is Kummer's confluent hypergeometric function.

Characteristic functionEdit

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:




Kurtosis excess:  


The entropy is given by:


where   is the polygamma function.

Related distributionsEdit

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

See alsoEdit

External linksEdit