Chi Distribution

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What is the chi distribution?

The chi distribution is an asymmetric continuous probability distribution over the non-negative real line. It is the continuous distribution of a variable whose square root is the chi-square distribution. Equivalently, the distribution can be thought of as the distribution of Euclidean distances of random variables from the origin.

One practical use of this 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]. It is also widely used in hypothesis testing and power analysis.

The Rayleigh distribution and the Maxwell–Boltzmann distribution (used in chemistry to describe the distribution of the speeds of molecules in an ideal gas) are two of the most familiar examples of the chi distribution, which itself is a special case of the noncentral chi-square distribution.

Chi distribution properties

chi distribution for various degrees of freedom cc0
The chi distribution for various degrees of freedom (k).

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.

With df = n > 0 degrees of freedom, the probability density function (PDF) is:

chi distribution
where x is positive and Γ is the gamma function.

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:


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

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

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