Probability Density Function of F-Distribution/Formulation 2

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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:

$\map {f_X} x = \dfrac {\map \Gamma {\dfrac {n + m} 2} n^{n / 2} m^{m / 2} } {\map \Gamma {n / 2} \map \Gamma {m / 2} } \ds \int_0^x t^{\paren {n / 2} - 1} \paren {m + n t}^{-\paren {n + m} / 2} \rd t$

where $\Gamma$ denotes the gamma function.


Proof


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Sources

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