Category:Probability Density Function of F-Distribution

This category contains pages concerning Probability Density Function of F-Distribution:

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.