Page 255 - Probability and Statistical Inference

232 4. Functions of Random Variables and Sampling Distribution

4.3.4 Consider the two related questions.
(i) Suppose that a random variable U has its pdf given by g(u) =
1
-σe –|u|/σ I(–∞ < u < ∞). First show that the mgf of U is given by
2
–1
2 2 –1
(1– σ t ) for |t|< σ , with 0 < σ < ∞;
(ii) Next, suppose that X , X are iid with the common pdf f(x) ,
1 2
with β > 0. Using part (i), derive the pdf
of V = X – X .
1
2
4.3.5 Let Y , Y be iid N (0, 4). Use the mgf technique to derive the distri-
1
2
bution of . Then, obtain the exact algebraic expressions for
(i) P(U > 1);
(ii) P(1 < U < 2);
(iii) P(| U – 2 |≥ 2.3).
4.3.6 Suppose that Z is the standard normal variable. Derive the expres-
sion of E[exp(tZ )] directly by integration. Hence, obtain the distribution of
2
2
Z .
4.3.7 (Exercise 4.3.6 Continued) Suppose that the random variable X is
distributed as N(µ, σ ), µ ∈ ℜ, σ ∈ ℜ . Obtain the expression of the mgf of
+
2
2
the random variable X , that is E[exp(tX )].
2
4.3.8 Prove the Theorem 4.3.2.
4.3.9 Suppose that U , ..., U are iid random variables with the common
1 n
pdf given by . Then, use the
part (ii) or (iii) from the Theorem 4.3.2 to obtain the distribution of the ran-
dom variable .
4.4.1 Let X be a random variable with the pdf

Find the marginal pdf’s of U, V, and W defined as follows:
2
(i) U = 2X – 1; (ii) V = X ; (iii)
4.4.2 Suppose that a random variable X has the Raleigh density given by

2
Derive the pdf of U = X .
4.4.3 (Example 4.4.3 Continued) Suppose that Z has the standard nor-
mal distribution. Let us denote Y = |Z|. Find the pdf of Y. {Caution: The