# F - Distribution

Consider two independent ${\displaystyle \chi ^{2}}$ random variables ${\displaystyle Y_{1}{\text{ and }}Y_{2}}$, with ${\displaystyle f_{1}{\text{ a }}f_{2}}$ degrees of freedom respectively, then the random variable: ${\displaystyle {X={\frac {\frac {Y_{1}}{f_{1}}}{\frac {Y_{2}}{f_{2}}}}}}$ will have a F-distribution (denoted as ${\displaystyle F(f_{1},f_{2})}$)with parameters ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ . The ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ parameters represent the degrees of freedom for the ${\displaystyle \chi ^{2}}$ distributed random variables in the numerator and the denominator. An F-distribution with parameters ${\displaystyle f_{1}{\text{ and }}f_{2}}$, has expected value and variance ${\displaystyle E(X)={\frac {f_{2}}{f_{2}-2}},\quad {\text{for }}{f_{2}>2}}$ ${\displaystyle Var(X)={\frac {2f_{2}^{2}(f_{1}+f_{2}-2)}{f_{1}(f_{2}-2)^{2}(f_{2}-4)}},\quad \ {\text{for}}\ f_{2}>4}$ This diagram plots some F-distributions for different values of ${\displaystyle f_{1}{\text{ and }}f_{2}}$.
The Chi-square, t-, and F- distributions are distributions that are functions of Normal random variables that are particularly useful in statistics. On the F-Distribution. The density function of an F-distribution is right-skewed. Increasing the values of ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ reduces this skewness. As ${\displaystyle f_{1}\rightarrow \infty }$ and ${\displaystyle f_{2}\rightarrow \infty }$, the density of the F-distribution will tend to a standard Normal distribution. The F-distribution is plotted for different values of ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$.