
4 Functions of Random Variables
Section 4.1: Exact Functions of One Variable
4.1. Derive the probability density function for Y, given that Y = X
4
, for the following densities. Sketch
all density functions.
(a) f
X
(x) =
1
2
, 0 < x < 2
(b) f
X
(x) = c exp(−x), 0 < x < ∞.
Solution: For any density function we need to use the following:
g(x) = X
4
g
(x) = 4X
3
.
Mathematically there are four roots (obtained by defining Y = Z
2
and Z = X
2
and performing successive
square roots) given by
Y = Z
2
→ Z
1,2
= ±
√
Y
Z = X
2
→ X
1,2
= ±
√
Z = ±
*
±
√
Y .
For a particular realization y then, the four roots are
x
1
=
4
√
y, x
2
= −
4
√
y,
x
3
= i
4
√
y, x
4
= −i
4
√
y.
In general then,
f
Y
(y) =
4
i=1
f
X
(x
i
)
|g
(x
i
)|
.
(a)