F Distribution

Definition

The pdf for the F distribution is

$$y=f\left(x|{\nu }_1,{\nu }_2\right)=\frac{\Gamma \left[\frac{\left({\nu }_1+{\nu }_2\right)}{2}\right]}{\Gamma \left(\frac{{\nu }_1}{2}\right)\Gamma \left(\frac{{\nu }_2}{2}\right)}{\left(\frac{{\nu }_1}{{\nu }_2}\right)}^{\frac{{\nu }_1}{2}}\frac{x^{\frac{{\nu }_1-2}{2}}}{{\left[1+\left(\frac{{\nu }_1}{{\nu }_2}\right)x\right]}^{\frac{{\nu }_1+{\nu }_2}{2}}}$$

for x ≥ 0, where Γ( · ) is the Gamma function.

Background

The F distribution has the following relationship with the chi-square distribution. If χ₁ and χ₂ are both independent and chi-square distributed with ν₁ and ν₂ degrees of freedom respectively, then the statistic F below is F-distributed.

$$F\left({\nu }_1,{\nu }_2\right)=\frac{\raisebox{1ex}{${\chi }_1$}\!\left/ \!\raisebox{-1ex}{${\nu }_1$}\right.}{\raisebox{1ex}{${\chi }_2$}\!\left/ \!\raisebox{-1ex}{${\nu }_2$}\right.}$$

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 χ₁ and χ₂, respectively.

Examples

Compute and Plot F Probability Density Function

Compute the probability density function (pdf) of the F distribution with 3 numerator degrees of freedom and 5 denominator degrees of freedom, over the range [0,5].

x = 0:0.01:5;
p = fpdf(x,3,5);

Plot the pdf.

figure;
plot(x,p)
grid on
xlabel("x")
ylabel("p")