chapter 13

6 Pages

## - F Distribution

Let X and Y be independent chi-square random variables with degrees of freedoms (dfs) m and n, respectively. The distribution of the ratio

Fm,n = (X m )

(Y n )

is called the F distribution with the numerator df = m and the denominator df = n. The probability density function of an Fm,n distribution is given by

f(x|m,n) = Î“ ( m+n 2

) Î“ ( m 2

) Î“ ( n 2

)m/2 [ 1 + mx

]m/2+n/2 , m > 0, n > 0, x > 0. Let S2i denote the variance of a random sample of size ni from a N(Âµi, Ïƒ

2) distribution, i = 1, 2. Then the variance ratio

distribution. For this reason, the F distribution is also known as the variance ratio distribution.