
3.6. Random variable X is continuous and its values are governed by f
X
(x) or F
X
(x). If random variable
Y = 2X, derive an expression for f
Y
(y) and F
Y
(y).
Solution: Consider a simple case where f
X
(x) = 1 for 0 ≤ x < 1. The corresponding cumulative distribution
is F
X
(x) = x for 0 ≤ x < 1 and F
X
(x) = 1 for x ≥ 1. We wish to use another random variable, Y, that is
related to our original random variable, X, by Y = 2X.
The probability does not change even when the random variables change. That is,
F
y
(y) = F
X
(2x) .
For example, Pr (x < 1/2) = 1/2 = F
X
(x = 1/2) = F
Y
(y = 2x = 1) . We apply the same principle for the
incremental probability,
Pr (x ≤ X ≤ x +