Sum of squared standard normals
We now generalise from the distribution of a single squared \(\NormalDistn(0, 1)\) variable to the sum of squares of \(k\) independent ones.
Definition
If \(\{Z_1, Z_2, \dots, Z_k\}\) are independent variables with standard normal distributions, then \(Y = \sum_{i=1}^k {Z_i^2}\) is said to have a Chi-squared distribution with \(k\) degrees of freedom,
\[ Y \;\;\sim\;\; \ChiSqrDistn(k\;\text{df}) \]Since z-scores have a standard normal distribution, this means that
\[ \sum_{i=1}^k \frac{(X_i - \mu)^2}{\sigma^2} \;\;\sim\;\; \ChiSqrDistn(k\;\text{df}) \]The properties of \(Y\) can be found by noting that
Relationship to the Gamma distribution
The \(\ChiSqrDistn(k\;\text{df})\) distribution is identical to a \(\GammaDistn(\frac k 2, \frac 1 2)\) distribution
\[ \ChiSqrDistn(k\;\text{df}) \;\;\equiv\;\; \GammaDistn(\frac k 2, \frac 1 2) \](Proved in full version)
The Chi-squared distribution's pdf can be found directly from that of the Gamma distribution.
Probability density function
A random variable \(Y \sim \ChiSqrDistn(k\;\text{df})\) has probability density function
\[ f(y) \;=\; \frac 1 {\Gamma({\large\frac k 2}) 2^{\large\frac k 2}} y^{\large{\frac k 2} - 1}e^{\large{-\frac y 2}} \qquad\text{if } y \gt 0 \](Proved in full version)