PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Theorem. 1.3.2
Suppose X is a continuous RV with PDF f X (x) and let Y = h(X), where h(x) is
differentiable and monotonic, i.e., either strictly increasing or strictly decreasing.
Then the PDF of Y is given by:
Proof. Assume h is increasing. Then
f Y (y) = f X {h −1 (y)}|h −1 (y) ′ |
F Y (y) = P (Y ≤ y) = P {h(X) ≤ y}
= P {X ≤ h −1 (y)}
= F X {h −1 (y)}
Figure 10: h increasing.
⇒ f Y (y) = d
dy F Y (y) = d
dy F X{h −1 (y)}
(use Chain Rule)
= f X
{
h −1 (y) } h −1 (y) ′ . (1)
Now consider the case of h(x) decreasing:
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