chi-squared random variable

A central chi-squared random variable X with n>0 degrees of freedom is given by the probability density functionMathworldPlanetmath


for x>0, where Γ represents the gamma functionDlmfDlmfMathworldPlanetmath.

The parameterMathworldPlanetmath n is usually, but not always, an integer, in which case the distributionDlmfPlanetmath is that of the sum of the squares of a sequence of n independent standard normal variables ( X1,X2,,Xn,


Parameters: n(0,).

Syntax: Xχ(n)2

Figure 1: Densities of the chi-squared distribution for different degrees of freedom.


  1. 1.

    This distribution is very widely used in statisticsMathworldMathworldPlanetmath, such as in hypothesis tests and confidence intervals.

  2. 2.

    The chi-squared distribution with n degrees of freedom is a result of evaluating the gamma distributionMathworldPlanetmath with α=n2 and λ=12.

  3. 3.


  4. 4.


  5. 5.

    The moment generating function is


    and is defined for all t with real partDlmfPlanetmath ( less than 1/2.

  6. 6.

    The sum of independent χ(m)2 and χ(n)2 random variablesMathworldPlanetmath has the χ(m+n)2 distribution.

Title chi-squared random variable
Canonical name ChisquaredRandomVariable
Date of creation 2013-03-22 11:54:49
Last modified on 2013-03-22 11:54:49
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Definition
Classification msc 60-00
Classification msc 11-00
Classification msc 20-01
Classification msc 20A05
Synonym central chi-squared distribution
Related topic ChiSquaredStatistic