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 [1].
The F-statistic
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.
Reference
- Feller W., An Introduction to Probability Theory and Its Applications, Vol II, Second Edition, John Wiley & Sons (1971)