Page 214 - Elements of Distribution Theory

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200 Distribution Theory for Functions of Random Variables

yielding the distribution of Y.For instance, if X has a discrete distribution with frequency
function p X , then the distribution of Y is discrete with frequency function

p Y (y) = p X (x).
x∈X: g(x)=y
However, in more general cases, difficulties may arise when attempting to implement this
approach. For instance, the set {x ∈ X:g(x) ∈ A} is often complicated, making computation
of the integral in (7.1) difficult. The analysis is simplified if g is a one-to-one function.

Theorem 7.1. Let X denote a real-valued random variable with range X and distribution
function F X . Suppose that Y = g(X) where g is a one-to-one function on X. Let Y = g(X)
and let h denote the inverse of g.
(i) Let F Y denote the distribution function of Y. If g is an increasing function, then

F Y (y) = F X (h(y)), y ∈ Y
If g is a decreasing function, then
F Y (y) = 1 − F X (h(y)−), y ∈ Y.

(ii) If X has a discrete distribution with frequency function p X , then Y has a discrete
distribution with frequency function p Y , where

p Y (y) = p X (h(y)), y ∈ Y.

(iii) Suppose that X has an absolutely continuous distribution with density function p X
and let g denote a continuously differentiable function. Assume that there exists

an open subset X 0 ⊂ X with Pr(X ∈ X 0 ) = 1 such that |g (x)| > 0 for all x ∈ X 0
and let Y 0 = g(X 0 ). Then Y has an absolutely continuous distribution with density
function p Y , where

p Y (y) = p X (h(y))|h (y)|, y ∈ Y 0 .
Proof. If g is increasing on X, then, for y ∈ Y,

F Y (y) = Pr(Y ≤ y) = Pr(X ≤ h(y)) = F X (h(y)).
Similarly, if g is decreasing on X, then

Pr(Y ≤ y) = Pr(X ≥ h(y)) = 1 − Pr(X < h(y)) = 1 − F(h(y)−).
Part (ii) follows from the fact that, for a one-to-one function g,

Pr(Y = y) = Pr(X = g(y)).
Consider a bounded function f defined on Y 0 = g(X 0 ). Since Pr(X ∈ X 0 ) = 1,

E[ f (Y)] = f (g(x))p X (x) dx.
X 0
By the change-of-variable formula for integration,

E[ f (Y)] = f (y)p X (h(y))|h (y)| dy.
Y 0