Page 160 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 153

4.51. Find the moment generating function of a normal r.v. N(p; a2).
If X is N(0; I), then from Prob. 4.1 (or Prob. 4.37), we see that Y = oX + p is N(p; 02), Then by setting
a = a and b = p in Eq. (4.120) (Prob. 4.50) and using Eq. (4.1 l9), we get

4.52. Let XI, . . . , X, be n independent r.v.'s and let the moment generating function of Xi be Mxi(t).
Let Y = X, + . + X,. Find the moment generating function of Y.
By definition (4.40),

4.53. Show that if XI, . . . , X, are independent Bernoulli r.v.'s with the parameter p, then Y = X, +
. + X, is a binomial r.v. with the parameters (n, p).
Using Eqs. (4.122) and (4.1 15), the moment generating function of Y is

which is the moment generating function of a binomial r.v. with parameters (n, p) [Eq. (4.1 l6)J. Hence, Y is
a binomial r.v. with parameters (n, p).

4.54. Show that if XI, . . . , X, are independent Poisson r.v.'s Xi having parameter Ai, then Y = X, +
. . + X, is also a Poisson r.v. with parameter 3, = A, + . + A,.
Using Eqs. (4.1 22) and (4.1 17), the moment generating function of Y is

which is the moment generating function of a Poisson r.v. with parameter 1. Hence, Y is a Poisson r.v. with
parameter 1 = Xti = 1, + + 1,.
Note that Prob. 4.15 is a special case for n = 2.

4.55. Show that if XI, . . . , X, are independent normal r.v.'s and Xi = N(pi; ai2), then Y = XI +
. . + X, is also a normal r.v. with mean p = p1 + . . + p, and variance a2 = aI2 + - . + an2.
Using Eqs. (4.1 22) and (4.1 21), the moment generating function of Y is

which is the moment generating function of a normal r.v. with mean p and variance a2. Hence, Y is a
normal r.v. with mean p = p, + - + p, and variance 02 = a12 + - + on2.
Note that Prob. 4.18 is a special case for n = 2 with pi = 0 and ai2 = 1.

4.56. Find the moment generating function of a gamma r.v. Y with parameters (n, A).
.
.
From Prob. 4.33, we see that if XI, . , X, are independent exponential r.v.'s, each with parameter A,
then Y = X, + . . + Xn is a gamma r.v. with parameters (n, A). Thus, by Eqs. (4.122) and (4.118), the
moment generating function of Y is