# Definition:F-Distribution

## Definition

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\Img X = \hointr 0 \infty$.

Let $n, m$ be strictly positive integers.

$X$ is said to have an F-distribution with $\tuple {n, m}$ degrees of freedom if and only if it has probability density function:

$\displaystyle \map {f_X} x = \frac {m^{m / 2} n^{n / 2} x^{\paren {n / 2} - 1} } {\paren {m + n x}^{\paren {n + m} / 2} \map \Beta {n / 2, m / 2} }$

where $\Beta$ denotes the beta function.

This is written:

$X \sim F_{n, m}$

## Also see

• Results about the F-distribution can be found here.