# Documentation

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# chi2pdf

Chi-square probability density function

Y = chi2pdf(X,V)

## Description

Y = chi2pdf(X,V) computes the chi-square pdf at each of the values in X using the corresponding degrees of freedom in V. X and V can be vectors, matrices, or multidimensional arrays that have the same size, which is also the size of the output Y. A scalar input is expanded to a constant array with the same dimensions as the other input. The degrees of freedom parameters in V must be positive, and the values in X must lie on the interval [0 Inf].

The chi-square pdf for a given value x and ν degrees of freedom is

$y=f\left(x|\nu \right)=\frac{{x}^{\left(\nu -2\right)/2}{e}^{-x/2}}{{2}^{\nu /2}\Gamma \left(\nu /2\right)}$

where Γ( · ) is the Gamma function.

If x is standard normal, then x2 is distributed chi-square with one degree of freedom. If x1, x2, ..., xn are n independent standard normal observations, then the sum of the squares of the x's is distributed chi-square with n degrees of freedom (and is equivalent to the gamma density function with parameters ν/2 and 2).

## Examples

nu = 1:6;
x = nu;
y = chi2pdf(x,nu)
y =
0.2420  0.1839  0.1542  0.1353  0.1220  0.1120

The mean of the chi-square distribution is the value of the degrees of freedom parameter, nu. The above example shows that the probability density of the mean falls as nu increases.