Like the t distribution, the F distribution is difficult to manipulate mathematically, so we only state the formulae for its mean and variance without proof.
Mean and variance
If \(F \sim \FDistn(k_1,\; k_2 \text{ df})\), then
\[ E[F] \;=\; \begin{cases} \dfrac {k_2}{k_2-2} & \quad\text{if } k_2 > 2 \\[0.4em] \infty & \quad\text{otherwise} \end{cases} \]and
\[ \Var(F) \;=\; \begin{cases} 2\left(\dfrac {k_2}{k_2-2}\right)^2 \dfrac{k_1 + k_2 - 2}{k_1(k_2 - 4)} & \quad\text{if } k_2 > 4 \\[0.4em] \infty & \quad\text{otherwise} \end{cases} \]Note that
Shape of the distribution
The F distribution is non-negative and skew with a long positive tail. The skewness is most extreme when \(k_2\) is small and can even result in an infinite mean and variance. The following diagram shows the shape of the distribution for some values of its degrees of freedom.
Illustration of distribution's shape
The diagram below shows the shape of the F distribution for various different values of its degrees of freedom.
Drag the sliders to see how the shape of the F distribution depends on the degrees of freedom. Note that
In most practical applications of the F distribution, the denominator's degrees of freedom are reasonably large.