Chi Distribution

< List of probability distributions

The chi distribution is the continuous distribution of a variable whose square root is the chi-squared distribution. Equivalently, it can be thought of as the distribution of Euclidean distances of the random variable from the origin. The two distributions are often confused, yet they are quite different — although the chi square distribution is used more often.

If a random variable X has a chi-square distribution with n ∈(0,∞) degrees of freedom, then U=√X is a chi distribution with n degrees of freedom. If the random variable is drawn from a noncentral chi-square distribution, then the distribution is called a noncentral chi distribution.

The chi distribution for various degrees of freedom (k).

One practical use of the chi distribution is to model the sample standard deviation for samples drawn from a normal distribution; that’s because the sample variance for such samples follows a chi-square distribution [1].

Chi Distribution Properties

The chi distribution describes the square root of a variable distributed according to a chi-square distribution.; with df = n > 0 degrees of freedom has a probability density function (PDF) of:

chi distribution
Probability density function: Γ is the gamma function

For positive values of x.

The cumulative distribution function (CDF) for this function does not have a closed form, but it can be approximated with a series of integrals, using calculus [2].

Expected value:

Expected value. Γ is the gamma function

This distribution is a special case of the generalized gamma distribution, or the Nakagami distribution.


[1] Abell, M. et. all. (1999). Statistics with Mathematica. Elsevier Science.

[2] UGC NET Education Paper II Chapter Wise Notebook | Complete Preparation Guide