Definition:Chi-Squared Distribution

Definition

Let $r$ be a strictly positive integer.

Let $X_1, X_2, \ldots, X_r$ be $r$ pairwise independent continuous random variables each with a standard Gaussian distribution.

Let $X := \ds \sum_{i \mathop = 1}^r {X_i}^2$ be the sum of the squares of $X_1, X_2, \ldots, X_r$.

Then $X$ is said to have a chi-squared distribution with $r$ degrees of freedom.

This is written:

$X \sim \chi_r^2$

where $\chi$ is the Greek letter $\chi$ (chi).

Also see

• Definition:Gamma Distribution Family, of which the chi-squared distribution is a member

• Results about the chi-squared distribution can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chi-squared distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chi-squared distribution

• Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Chi-SquaredDistribution.html