Probability Density Function of F-Distribution/Formulation 1

Theorem

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.

Let $X$ have an $F$-distribution with $\tuple {n, m}$ degrees of freedom.

The probability density function of $X$ is:

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

where $\Beta$ denotes the beta function.

Proof

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Sources

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