A simple transformation of a standard normal random variable has a distribution called a chi-squared distribution.

Definition

If a random variable, \(Z\), has a standard normal distribution,

\[ Z \;\;\sim\;\; \NormalDistn(0, 1) \]

then we say that its square, \(Y = Z^2\), has a chi-squared distribution with 1 degree of freedom,

\[ Y \;\;\sim\;\; \ChiSqrDistn(1\;\text{df}) \]

This also means that

\[ \frac{(X - \mu)^2}{\sigma^2} \;\;\sim\;\; \ChiSqrDistn(1\;\text{df}) \]

Probability density function

The random variable \(Y \sim \ChiSqrDistn(1\;\text{df})\) has probability density function

\[ f(y) \;=\; \frac 1 {\sqrt{2\pi}} y^{\large{-\frac 1 2}}e^{\large{-\frac y 2}} \qquad\text{if } y \gt 0 \]

(Proved in full version)

The Chi-squared distribution is actually a special case from the family of Gamma distributions.

Relationship to gamma distribution

The \(\ChiSqrDistn(1\;\text{df})\) distribution is identical to a \(\GammaDistn(\frac 1 2, \frac 1 2)\) distribution.

(Proved in full version)