
4.6. The random variables X and Y are related by the equation,
Y = e
X
.
(a) If X is uniformly distributed, 0 ≤ X ≤ 5, find f
Y
(y) . Sketch this density function.
(b) If X is governed by the density function f
X
= ce
−x
, 0 ≤ X ≤ 5, where c is a constant, find f
Y
(y) .
Sketch this density function.
Solution: (a) From the given information f
X
(x) = 1/5. Performing the density transformation (which is
valid for any density function of X):
x
1
= ln y
g(x) = exp(x)
g
(x) = exp(x)
f
Y
(y) =
f
X
(x
1
)
|g
(x
1
)|
=
1/5
exp(ln y)
=
1
5y
, 1 ≤ y ≤ 148.4
f
Y
(1) = 1/5
f
Y
(148.4) = 1/(5 · 148.4) = 1.3477 × 10
−3
.
The density functions are shown in the two figures given next.
Density function f
X
(x)