Let X be the standard normal distribution (with mean 0 and standard deviation 1). Then X² has the special distribution, which is usually referred to as the chi-square distribution. It is often denoted by χ² distribution, where χ is the lowercase Greek letter chi. A random variable X has the chi-square distribution with ν degrees of freedom if its probability density is given by

\[ f(x) = \begin{cases} \dfrac{1}{2^{\nu /2} \Gamma (\nu /2)} \, x^{\nu /2 -1} \, e^{-x/2} , & \quad\mbox{for } x>0, \\ 0, & \quad\mbox{elsewhere}. \end{cases} \]

The mean and the variance of the chi-square distribution with ν degrees of freedom are ν and 2ν, respectively.

We built an example of chi-square distribution from three standard normal distributions.

R code

You can compare chi-square distributions with three and four degrees of freedom:

R code

We can generate chi-square distribution with, say, 20 degrees of freedom directly using the following R script:

R code