
4.3. Derive the probability density function for Y, given Y = a + X
3
. Consider two cases:
(a) f
X
(x) = c exp(−x), x ≥ 0,
(b) f
X
(x) is a lognormal density.
Sketch all density functions.
Solution: (a) Given the density f
X
(x) = c exp(−x), x ≥ 0, first find the value of c :
∞
0
c exp(−x)dx = 1 =⇒ c = 1.
For
X
3
= Y −a,
X has three roots, and the only real root is
X =
3
√
Y − a.
Performing the density transformation:
x
1
=
3
√
y −a, x
2
=
3
√
y −a, x
3
=
3
√
y − a
g(x) = a + X
3
g
(x) = 2X
2
f
Y
(y) =
f
X
(x
1
)
|g
(x
1
)|
+
f
X
(x
2
)
|g
(x
2
)|
+
f
X
(x
3
)
|g
(x
3
)|
=
3
2(y −a)
2
3
exp(−
3
√
y −a), y ≥ 0, y = a.
Density functions for a = 1 are shown in the two figures below.
Density Function f
X
(x) = exp(−x), x > 0
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