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 also means that if \(\{X_1, X_2, \dots, X_k\}\) are a random sample from a \(\NormalDistn(\mu, \sigma^2)\), distribution, then
\[ \sum_{i=1}^k \frac{(X_i - \mu)^2}{\sigma^2} \;\;\sim\;\; \ChiSqrDistn(k\;\text{df}) \]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
\[ \ChiSqrDistn(k\;\text{df}) \;\;\equiv\;\; \GammaDistn(\frac k 2, \frac 1 2) \]We stated earlier that if \(Y_1 \sim \GammaDistn(\alpha_1,\; \beta)\) and \(Y_2 \sim \GammaDistn(\alpha_2,\; \beta)\) are independent, then
\[ Y_1 + Y_2 \;\;\sim\;\; \GammaDistn(\alpha_1 + \alpha_2,\; \beta) \]This can be easily generalised to the sum of \(k\) independent \(\GammaDistn(\alpha_i, \beta)\) variables:
\[ \sum_{i=1}^{k} {Y_i} \;\;\sim\;\; \GammaDistn(\sum_{i=1}^{k} {\alpha_i},\; \beta) \]Since a Chi-squared random variable \(Y\) is the sum of \(k\) independent \(\GammaDistn(\frac 1 2, \frac 1 2)\) variables, it therefore also has a Gamma distribution,
\[ \ChiSqrDistn(k\;\text{df}) \;\;\equiv\;\; \GammaDistn(\frac k 2, \frac 1 2) \]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
\[ f(y) \;=\; \frac 1 {\Gamma({\large\frac k 2}) 2^{\large\frac k 2}} y^{\large{\frac k 2} - 1}e^{\large{-\frac y 2}} \qquad\text{if } y \gt 0 \]Replacing \(\alpha = \frac k 2\) and \(\beta = \frac 1 2\) in the \(\GammaDistn(\frac k 2, \frac 1 2)\) distribution's pdf gives the pdf of the Chi-squared distribution.