The F distribution's mean and variance are stated below 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

\[ \Var(F) \;=\; \begin{cases} 2\left(\dfrac {k_2}{k_2-2}\right)^2 \dfrac{k_1 + k_2 - 2}{k_1(k_2 - 4)} & \quad\text{if } k_2 > 4 \\[0.4em] \infty & \quad\text{otherwise} \end{cases} \]

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 diagram below shows its shape for a few values of the degrees of freedom.