Probability Density Function of F-Distribution
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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.
Formulation 1
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.
Formulation 2
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
- Weisstein, Eric W. "F-distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/F-Distribution.html