Page 215 - Elements of Distribution Theory

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7.2 Functions of a Real-Valued Random Variable 201

It follows that the distribution of Y is absolutely continuous with density function

p X (h(y))|h (y)|, y ∈ Y 0 ;
part (iii) follows.

Note that in applying part (iii) of Theorem 7.1, often the set X 0 may be taken to be X.
Also note that parts (i) and (ii) of the theorem give F Y (y) only for y ∈ Y;in typical cases,
values of F Y (y) for y ∈ Y can be obtained by inspection.

Example 7.2 (Lognormal distribution). Let X denote a random variable with a normal
distribution with mean µ and standard deviation σ. Let Y = exp(X)so that log Y has a
normal distribution; the distribution of Y is known as a lognormal distribution.
Recall that the distribution of X is absolutely continuous with density function
1 1 2
p X (x) = √ exp − (x − µ) , −∞ < x < ∞.
σ (2π) 2σ 2

We may write Y = g(X) with g(x) = exp(x); then g (x) = exp(x) > 0 for all x ∈ R and
g has inverse h(y) = log(y). Hence, X 0 = X = R and Y 0 = (0, ∞). It follows that the
distribution of Y is absolutely continuous with density
1 1 1 2
p Y (y) = √ exp − (log(y) − µ) , y > 0.
y σ (2π) 2σ 2

Example 7.3 (Empirical odds ratio). Let X denote a random variable with a binomial
distribution with parameters n and θ; then the distribution of X is discrete with frequency
function

n x n−x
p X (x) = θ (1 − θ) , x = 0,..., n.
x
Let
X + 1/2
Y = ;
n − X + 1/2
hence, if X denotes the number of successes in n trials, Y denotes a form of the empirical
odds ratio based on those trials. The function g is given by g(x) = (x + 1/2)/(n − x + 1/2)
with inverse
(n + 1/2)y − 1/2
h(y) = .
1 + y
It follows that the distribution of Y is discrete with frequency function

n h(y) h(y)

p Y (y) = θ (1 − θ)
h(y)
for values of y in the set
1 3 5

g(X) = , , ,..., 2n + 1 .
2n + 1 2n − 1 2n − 3