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)