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192 4. Functions of Random Variables and Sampling Distribution

distributed as N(µ , ), i = 1, ..., n. With fixed but otherwise arbitrary real
i
numbers a , ..., a , we write and then denoting
1 n
and we claim along the lines of the Example 4.3.3 that the sam-
2
pling distribution of U turns out to be N(µ, σ ). It is left as the Exercise 4.3.3.!
For the record, we now state the following results. Each part has already
been verified in one form or another. It will, however, be instructive to supply
direct proofs using the mgf technique under these special situations. These
are left as the Exercise 4.3.8.
Theorem 4.3.2 (Reproductive Property of Independent Normal,
Gamma and Chi-square Distributions) Let X , ..., X be independent ran-
n
1
dom variables. Write and = n U. Then, one can conclude the
–1
following:
(i) If X ’s have the common N(µ, σ ) distribution, then U is distributed
2
i
as N(nµ, nσ ) and hence is distributed as N(µ, 1/nσ );
2
2
(ii) If X ’s have the common Gamma (α, β) distribution, then U is dis
i
tributed as Gamma(nα, β);
(iii) If X has a Gamma(½ν , 2) distribution, that is a Chi-square distri
i i
bution with ν degrees of freedom for i = 1, ..., n, then U is distrib
i
uted as Gamma(½ν, 2) with which is a Chi-square
distribution with ν degrees of freedom.
Example 4.3.6 (Example 4.1.1 Continued) Now, let us briefly go back to
the Example 4.1.1. There, we had Z , Z iid N(0, 1) and in view of the Ex-
1
2
ample 4.2.6 we can claim that are iid . Thus, by using the repro-
ductive property of the independent Chi-squares, we note that the random
variable is distributed as the random variable. The pdf of
W happens to be g(w) = 1/2e –w/2 I(w > 0). Hence, in retrospect, (4.1.4) made
good sense. !

4.4 A General Approach with Transformations

This is a more elaborate methodology which can help us to derive distribu-
tions of functions of random variables. We state the following result without
giving its proof.
Theorem 4.4.1 Consider a real valued random variable X whose pdf is
f(x) at the point x belonging to some subinterval χ of the real line ℜ. Sup-
pose that we have a one-to-one function g: χ → ℜ and let g(x) be differ-
entiable with respect to x(∈ χ). Define the transformed random variable