Chi-square probability density function
Y = chi2pdf(X,V)
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 integers, 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
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).
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.