In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. Equivalently, it is the distribution of the Euclidean distance between a multivariate Gaussian random variable and the origin. The chi distribution describes the positive square roots of a variable obeying a chi-squared distribution.

chi
Probability density function
Plot of the Chi PMF
Cumulative distribution function
Plot of the Chi CMF
Notation or
Parameters (degrees of freedom)
Support
PDF
CDF
Mean
Median
Mode for
Variance
Skewness
Excess kurtosis
Entropy
MGF Complicated (see text)
CF Complicated (see text)

If are independent, normally distributed random variables with mean 0 and standard deviation 1, then the statistic

is distributed according to the chi distribution. The chi distribution has one positive integer parameter , which specifies the degrees of freedom (i.e. the number of random variables ).

The most familiar examples are the Rayleigh distribution (chi distribution with two degrees of freedom) and the Maxwell–Boltzmann distribution of the molecular speeds in an ideal gas (chi distribution with three degrees of freedom).

Definitions

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Probability density function

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The probability density function (pdf) of the chi-distribution is

 

where   is the gamma function.

Cumulative distribution function

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The cumulative distribution function is given by:

 

where   is the regularized gamma function.

Generating functions

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The moment-generating function is given by:

 

where   is Kummer's confluent hypergeometric function. The characteristic function is given by:

 

Properties

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Moments

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The raw moments are then given by:

 

where   is the gamma function. Thus 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:   which is close to   for large k.

Variance:   which approaches   as k increases.

Skewness:  

Kurtosis excess:  

Entropy

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The entropy is given by:

 

where   is the polygamma function.

Large n approximation

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We find the large n=k+1 approximation of the mean and variance of chi distribution. This has application e.g. in finding the distribution of standard deviation of a sample of normally distributed population, where n is the sample size.

The mean is then:

 

We use the Legendre duplication formula to write:

 ,

so that:

 

Using Stirling's approximation for Gamma function, we get the following expression for the mean:

 
 
 
 

And thus the variance is:

 
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  • If   then   (chi-squared distribution)
  •   (half-normal distribution), i.e. if   then  , and if   for any   then  
  •   (Rayleigh distribution) and if   for any   then  
  •   (Maxwell distribution) and if   for any   then  
  •  , the Euclidean norm of a standard normal random vector of with   dimensions, is distributed according to 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
  •   (Normal distribution)
  • The mean of the chi distribution (scaled by the square root of  ) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution.
Various chi and chi-squared distributions
Name Statistic
chi-squared distribution  
noncentral chi-squared distribution  
chi distribution  
noncentral chi distribution  

See also

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References

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  • Martha L. Abell, James P. Braselton, John Arthur Rafter, John A. Rafter, Statistics with Mathematica (1999), 237f.
  • Jan W. Gooch, Encyclopedic Dictionary of Polymers vol. 1 (2010), Appendix E, p. 972.
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