Chi-Square Distribution

Compact study note.

Summary

The chi-square distribution with $\nu$ degrees of freedom is the distribution of a sum of squared independent standard normal variables. It is right-skewed, especially for small $\nu$ .^[1]

Prerequisites

• Normal Distribution
• Gamma Distribution

Definition

If $Z_1,\dots ,Z_{\nu }$ are IID $N(0,1)$ , then $X=\sum _{i=1}^{\nu }{Z}_i^2\sim {\chi }_{\nu }^2$ .

Notation and Assumptions

$\nu$ is a positive degree-of-freedom parameter.

Parameters

$\nu >0$ in the gamma-family form; for the squared-normal construction, $\nu$ is a positive integer.

Support

$[0,\mathrm{\infty })$ .

PMF or PDF

$f_X(x)=\frac{1}{2^{\nu /2}\mathrm{\Gamma }(\nu /2)}{x}^{\nu /2-1}{e}^{-x/2}$ for $x>0$ .

CDF

The CDF is the gamma CDF with shape $\nu /2$ and scale $2$ .

Moments

$E[X]=\nu$ , $\mathrm{Var}(X)=2\nu$ , and $M_X(t)=(1-2t{)}^{-\nu /2}$ for $t<1/2$ .

Essential Result

${\chi }_{\nu }^2$ is $\mathrm{Gamma}(\nu /2,2)$ in shape-scale notation.

Small Example

If $Z_1,Z_2,Z_3$ are independent standard normals, then $Z_1^2+Z_2^2+Z_3^2\sim {\chi }_3^2$ .

Common Mistakes

• Calling chi-square symmetric.
• Defining it through cubes or unsquared normal quantities.

Connections

• Normal Distribution
• Gamma Distribution
• F Distribution

References