The F-Distribution

The F-distribution is a continuous probability distribution that arises as a ratio of two independent scaled chi-square random variables.

If $U \sim \chi^2_{d_1}$ and $V \sim \chi^2_{d_2}$ are independent, then the random variable

$$F = \dfrac{U/d_1}{V/d_2}$$

follows the F-distribution with $d_1$ and $d_2$ degrees of freedom, written

$$F \sim F(d_1, d_2).$$

We call $d_1$ the numerator degrees of freedom and $d_2$ the denominator degrees of freedom. The order matters: $F(d_1, d_2)$ is generally not the same distribution as $F(d_2, d_1)$.

Since $U, V \geq 0$, the F-distribution is supported on $x > 0$, and its density is right-skewed.

For example, if $U \sim \chi^2_4$ and $V \sim \chi^2_9$ are independent, then

$$\dfrac{U/4}{V/9} \sim F(4, 9).$$