F-Distribution¶

If \(\chi_{n}^{2}\) and \(\chi_{m}^{2}\) are two independent chi-squared distributions with \(n\) and \(m\) degrees of freedom respectively, then the variable \(F_{n,m}\) defined as

\[ \begin{aligned} F_{n,m} = \frac{\chi_{n}^{2}/n}{\chi_{m}^{2}/m} \end{aligned} \]

is said to have an F-distribution with n and m degrees of freedom.

For any \(\alpha\) between \(0\) and \(1\), we define \(F_{\alpha, n, m}\) as

\[ \begin{aligned} P(F_{n,m} \geq F_{\alpha, n, m}) = \alpha \end{aligned} \]

These values are available in standard tables for different combinations of \(\alpha, n\) and \(m\).

Consider

\[ \begin{aligned} \alpha &= P(\frac{\chi_{n}^{2}/n}{\chi_{m}^{2}/m} > F_{\alpha, n, m})\newline &= P(\frac{\chi_{m}^{2}/m}{\chi_{n}^{2}/n} < \frac{1}{F_{\alpha, n, m}})\newline &= 1 - P(\frac{\chi_{m}^{2}/m}{\chi_{n}^{2}/n} \geq \frac{1}{F_{\alpha, n, m}})\newline \text{or} \quad P(\frac{\chi_{m}^{2}/m}{\chi_{n}^{2}/n} \geq \frac{1}{F_{\alpha, n, m}}) &= 1 - \alpha\newline \text{but} \quad P(F_{m,n} \geq F_{1 - \alpha, m, n}) &= 1 - \alpha\newline \text{From the last two equations} \quad \frac{1}{F_{\alpha, n, m}} &= F_{1-\alpha, m, n} \end{aligned} \]

Mean and Variance¶

We state the following without proof for \(X \sim F(n,m)\)

\[ \begin{aligned} E[X] &= \frac{m}{m - 2} \: m > 2\newline Var(x) &= \frac{2 m^{2}(m + n - 2)}{n(m - 2)^{2}(m - 4)} \: m > 4 \end{aligned} \]