The F distribution is also closely associated with random samples from normal distributions.

Definition

If \(X_1\) and \(X_2\) have chi-squared distributions,

\[ X_1 \;\sim\; \ChiSqrDistn(k_1 \text{ df}) \spaced{and} X_2 \;\sim\; \ChiSqrDistn(k_2 \text{ df}) \]

then

\[ F \;\;=\;\; \frac{\diagfrac{X_1}{k_1}} {\diagfrac{X_2}{k_2}} \]

is said to have an F distribution with \(k_1\) and \(k_2\) degrees of freedom.

\[ F \;\;\sim\;\; \FDistn(k_1,\;k_2 \text{ df}) \]

Its pdf is complex and cannot be proved here.

Probability density function

If a random variable has an F distribution with \(k_1\) and \(k_2\) degrees of freedom then its probability density function is

\[ f(x) \;\;=\;\; \frac{\Gamma\left(\frac{k_1+k_2}{2}\right)}{\Gamma\left(\frac{k_1}{2}\right) \Gamma\left(\frac{k_2}{2}\right)} \left(\frac{k_1}{k_2}\right)^{\frac{k_1}{2}} x^{\frac{k_1}{2} - 1} \left(1 + \frac{k_1x}{k_2} \right)^{-\frac {k_1+k_2}{2}} \qquad\text{for }x > 0 \]

Probabilities and quantiles

Cumulative probabilities and quantiles of the F distribution must be evaluated numerically. In Excel, the probability of an \(\FDistn(\langle k_1 \rangle, \langle k_2 \rangle)\) distribution being less than a value \(\langle x \rangle\), \(P(X \lt \langle x \rangle)\) can be found with

=F.DIST(\(\langle x \rangle\), \(\langle k_1 \rangle\), \(\langle k_2 \rangle\), TRUE)

In a similar way, its \(\langle p \rangle\)'th quantile is

=F.INV(\(\langle p \rangle\), \(\langle k_1 \rangle\), \(\langle k_2 \rangle\))