PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
F Y (y) = P (Y ≤ y) = P {h(X) ≤ y}
= P {X ≥ h −1 (y)}
= 1 − F X {h −1 (y)}
⇒ f Y (y) = −f X {h −1 (y)}h −1 (y) ′ (2)
Figure 11: h decreasing.
Finally, observe that if h is increasing then h ′ (x) and hence h −1 (y) ′ must be positive.
Similarly for h decreasing, h −1 (y) ′ < 0. Hence (1) and (2) can be combined to give:
f Y (y) = f X {h −1 (y)}|h −1 (y) ′ |
Examples:
1. Discrete transformation of single RV
X ∼ P o(λ) and Y is X rounded to the nearest multiple of 10.
Possible values of Y are: 0, 10, 20, . . .
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