Chi distribution
Probability density function  
Cumulative distribution function  
Parameters  (degrees of freedom) 

Support  
CDF  
Mean  
Mode  for 
Variance  
Skewness  
Ex. kurtosis  
Entropy 

MGF  Complicated (see text) 
CF  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
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
 If then (chisquared distribution)
 (Normal distribution)
 If then
 If then (halfnormal distribution) for any
 (Rayleigh distribution)
 (Maxwell distribution)
 (The 2norm 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
Name  Statistic 

chisquared distribution  
noncentral chisquared distribution  
chi distribution  
noncentral chi distribution 