F Distribution

Definition

The pdf for the F distribution is

$y = f (x | ν_{1}, ν_{2}) = \frac{Γ [\frac{(ν_{1} + ν_{2})}{2}]}{Γ (\frac{ν_{1}}{2}) Γ (\frac{ν_{2}}{2})} {(\frac{ν_{1}}{ν_{2}})}^{\frac{ν_{1}}{2}} \frac{x^{\frac{ν_{1} - 2}{2}}}{{[1 + (\frac{ν_{1}}{ν_{2}}) x]}^{\frac{ν_{1} + ν_{2}}{2}}}$

where Γ( · ) is the Gamma function.

Background

The F distribution has the following relationship with the chi-square distribution. If χ₁and χ₂ are both chi-square with ν₁ and ν₂ degrees of freedom respectively, then the statistic F below is F-distributed.

$F (ν_{1}, ν_{2}) = \frac{\frac{χ_{1}}{ν_{1}}}{\frac{χ_{2}}{ν_{2}}}$

The two parameters, ν₁ and ν₂, are the numerator and denominator degrees of freedom. That is, ν₁ and ν₂ are the number of independent pieces of information used to calculate χ₁and χ₂,respectively.