There is an important connection between an F variable and chi-squared variables. If X and Y are independent chi-squared rv’s with n1 and n2 df, respectively, then the rv
is said to have an F distribution with (n1, n2) degrees of freedom.
The random variable has the chi-square distribution with degrees of freedom with probability density function
The moment generating function for is
The moment generating function of is
which is the moment generating function of a chi-square random variable with degrees of freedom.
Since and follow independently chi-square distribution with we have that has a chi-square distribution with degrees of freedom.
has a chi-square distribution with degrees of freedom.
has an F distribution with (n1, n2) degrees of freedom.
and are independent.
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