Some important distributions in statistical applications are of the form where and are independent and . When and where and have chi-square distributions with degrees of freedom and , respectively, the ratio has an F distribution and this ratio is called the F-statistic.
Let and be the density functions of the independent random variables and , respectively. Let and be the distribution functions of and , respectively. Let where . We have . Integrating over the region of , we obtain the following.
Taking the derivative, we obtain .
This above derivation of the density function is found in .
Note that a chi-square distribution with degrees of freedom (denoted by ) is a gamma distribution with parameters and where is a positive integer. Suppose and where and have chi-square distributions with and as degrees of freedom, respectively. The following are the pdfs of and :
In general, if where is a constant, then we have this relationship for the density functions between and : . Then the following are the density functions of and . Note that has a gamma distribution with parameters and . For , it is a gamma distribution with parameters and .
Using , the following is the derivation of the density function of :
Simplifying the above, we obtain:
The distributon derived above is said to be an F distribution with and degrees of freedom. The first parameter is the degrees of freedom in the numerator and the second parameter is the degrees of freedom in the denominator.
- Feller W., An Introduction to Probability Theory and Its Applications, Vol II, Second Edition, John Wiley & Sons (1971)