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ISyE 6739 — Summer 2009
Homework #4 (Covers Module 15) — Solutions
1. Suppose X∼Unif(1,3). Find the p.d.f. of Z=eX.
Hint: The c.d.f. of Zis
G(z) = Pr(Z≤z)
=Pr(eX≤z)
=Pr(X≤`n(z))
=Z`n(z)
1f(x)dx (if 1 ≤`n(z)≤3)
= (`n(z)−1)/2.
Now you can get the p.d.f.
g(z) = d
dz G(z) = (0 if z < e or z > e3
1
2zif e≤z≤e3♦
2. Suppose Xhas p.d.f. f(x) = 2xe−x2,x≥0. Find the distribution of Z=X2.
Hint: The c.d.f. of Zis
G(z) = Pr(Z≤z)
=Pr(X2≤z)
=Pr(−√z≤X≤√z)
=Pr(0 ≤X≤√z) (since X≥0)
=Z√z
02xe−x2dx
= 1 −e−z.
Thus, Zis Exp(1). ♦
3. Computer Exercises — Random Variate Generation (see original assignment)
