F - Distribution

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