# Definition:F-Distribution

Jump to navigation
Jump to search

## 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.

## Sources

- Weisstein, Eric W. "F-distribution." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/F-Distribution.html

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 39$: Probability Distributions: $F$ Distribution: $39.7$