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The pdf for the *F* distribution is

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

where Γ( · ) is the Gamma function.

The *F* distribution has a natural relationship with the
chi-square distribution. If *χ*_{1 }and
*χ*_{2} are both chi-square with
*ν*_{1} and
*ν*_{2} 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, *ν*_{1} and
*ν*_{2}, are the numerator and
denominator degrees of freedom. That is,
*ν*_{1} and
*ν*_{2} are the number of independent
pieces of information used to calculate *χ*_{1
}and
*χ*_{2},_{ }respectively.

Compute the pdf of an *F* distribution with 5 numerator degrees of freedom and 3 denominator degrees of freedom.

x = 0:0.01:10; y = fpdf(x,5,3);

Plot the pdf.

figure; plot(x,y)

The plot shows that the *F* distribution exists on positive real numbers and is skewed to the right.

`fcdf`

| `finv`

| `fpdf`

| `frnd`

| `fstat`

| `random`