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4. Functions of Random Variables and Sampling Distribution 227

4.9 Exercises and Complements
4.1.1 (Example 4.1.1 Continued) Suppose that Z , Z are iid standard
1
2
normal variables. Evaluate with some fixed but arbitrary b >
0. {Hint: Express the required probability as the double integral,

. Use the substitutions from (4.1.1)

to rewrite this as which will change to

. This last integral can then be re-

placed by
Also, look at the Exercise 4.5.7.}
4.2.1 (Example 4.2.1 Continued) Consider the two random variables X ,
1
X from the Example 4.2.1. Find the pmf’s of the following random variables
2
which are functions of X , X .
1 2

4.2.2 Let X , X be iid Poisson(λ), λ > 0. Derive the distribution of the
1
2
random variable U = X + X . Then, use mathematical induction to show that
1
2
has the Poisson(nλ) distribution if X , ..., X are iid Poisson(λ).
1
n
Next, evaluate P(X – X = v) explicitly when v = 0, 1, 2, 3. Proceeding this
2
1
way, is it possible to write down the pmf of the random variable V = X – X ?
1
2
{Hint: In the first part, follow along the Examples 4.2.2-4.2.3.}
4.2.3 In a Bernoulli experiment, suppose that X = number of trials needed to
1
observe the first success, and X = number of trials needed since the first
2
success to observe the second success, where the trials are assumed indepen-
dent having the common success probability p, 0 < p < 1. That is, X , X are
2
1
assumed iid having the Geometric(p) distribution defined earlier in (1.7.7). First,
find the distribution of the random variable U = X + X . Next, with k such iid
1
2
random variables X , ..., X , derive the pmf of the random variable
1 k
. The distribution of U is called Negative Binomial with param-
eters (k, p). A different parameterization was given in (1.7.9).
4.2.4 Let the pdf of X be with β > 0. Find the
pdf’s of the following random variables which are functions of X.
(i) U = X ;
2
(ii) V = X .
3