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4. Functions of Random Variables and Sampling Distribution 211

as F ν2, ν1 . That is, 1/F has a F distribution too. This feature may be intuitively
viewed as the “symmetry property” of the pdf h(u). The explicit form of the
pdf has not played any crucial role in this conclusion.
How about finding the moments of the F ν1, ν2 variable? The pdf h(u) is not
essential for deriving the moments of U. For any positive integer k, observe
that

–k
as long as E[Y ] is finite. We can split the expectation in (4.5.10) because X
and Y are assumed independent. But, it is clear that the expression in (4.5.10)
will lead to finite entities provided that appropriate negative moment of a Chi-
square variable exists. We had discussed similar matters for the gamma distri-
butions in (2.3.24)-(2.3.26).
By appealing to (2.3.26) and (4.5.10), we claim that for the F , variable
ν1 ν2
2
given in the Definition 4.5.2, we have E(U) finite if ν > 2, whereas E(U ) is
2
finite, that is V(U) is finite when ν > 4. One should verify the following
2
claims:
and also

Example 4.5.3 The Two-Sample Problem: Let X , ..., X be iid
i1
ini
i = 1, 2, and that the X ’s are independent of the X ’s. For n ≥ 2,
1j 2j i
we denote as in the
Example 4.5.2. Now, , (ii) they are also
independent, and hence in view of the Definition 4.5.2, the random variable U
2
= (S /σ ) ÷ (S /σ ) has the F distribution with degrees of freedom ν = n –
1
1
2
2
1
1
1, ν = n – 1. !
2 2
4.5.3 The Beta Distribution
Recall the random variable U mentioned in the Definition 4.5.2. With k =
k(ν ,ν ) = (ν /ν ) 1/2 ν 1 Γ((ν + ν )/2) {Γ(ν /2)Γ(ν /2)} , the pdf of U was
–1
2
1
2
2
1
1
2
1
for 0 < u < ∞. Now, let us define Z = [(ν /ν )U]/[1 + (ν /ν )U]. The
2
1
2
1
transformation from u → z is one-to-one and we have (ν /ν )u = z(1 –
1 2