Page 168 - Probability, Random Variables and Random Processes

160 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS [CHAP. 4

Consider a r.v. X with pdf

Find the moment generating function of X.

Ans. M ,(t) = e - " +
Let X = N(0; 1). Using the moment generating function of X, determine E(Xn).
n = I, 3, 5, ...
Ans. E(Xn) = {e . 3 . . . . .
(n - 1) n = 2, 4, 6, ...

Let X and Y be independent binomial r.v.'s with parameters (n, p) and (m, p), respectively. Let Z = X + Y.
What is the distribution of Z?
Hint: Use the moment generating functions.
Ans. Z is a binomial r.v. with parameters (n + m, p).

Let (X, Y) be a continuous bivariate r.v. with joint pdf
x>o,y>o
.fxu(x. Y) =
otherwise
(a) Find the joint moment function of X and Y.
(h) Find the joint moments m,, , m,,, and m, , .
1
Ans. (a) MXAt,, t,) = (h) m,, = 1, m,, = 1, m,, = 1
(1 - t,)(l - t2)
Let (X, Y) be a bivariate normal r.v. defined by Eq. (3.88). Find the joint moment generating function of X
and Y.

Let X,, . . . , X, be n independent r.v.'s and Xi > 0. Let

Show that for large n, the pdf of Y is approximately log-normal.
Hint: Take the natural logarithm of Y and use the central limit theorem and the result of Prob. 4.10.

Let Y = (X - A)/JA, where X is a Poisson r.v, with parameter A. Show that Y = N(0; 1) when 1 is suffi-
ciently large.
Hint: Find the moment generating function of Y and let 1 -, co.

Consider an experiment of tossing a fair coin 1000 times. Find the probability of obtaining more that 520
heads (a) by using formula (4.136), and (h) by formula (4.139).
Ans. (a) 0.1038 (h) 0.0974

The number of cars entering a parking lot is Poisson distributed with a rate of 100 cars per hour. Find the
time required for more than 200 cars to have entered the parking lot with probability 0.90 (a) by using
formula (4.1 do), and (h) by formula (4.1 43).

Ans. (a) 2.1 89 h (h) 2.1946 h