Page 158 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 15 1

4.45. Let X be a Bernoulli r.v.
(a) Find the moment generating function of X.
(b) Find the mean and variance of X.
(a) By definition (4.40) and Eq. (2.32),
Mx(t) = E(etX) = etxipx(xi)
1
= et(O)pX(O) + et(')p,(l) -5 (1 - p) + pet

Hence,

4.46. Let X be a binomial r.v. with parameters (n, p).
Find the moment generating function of X.
Find the mean and variance of X.
By definition (4.40) and Eq. (2.36), and letting q = 1 - p, we get

The first two derivatives of Mx(t) are
M;(t) = n(q + pet)"- 'pet
Mi(t) = n(q + pet)"- lpet + n(n - l)(q + pet)"-2(pet)2
Thus, by Eq. (4.42),

Hence,

4.47. Let X be a Poisson r.v. with parameter A.
Find the moment generating function of X.
Find the mean and variance of X.
By definition (4.40) and Eq. (2.40),
w 1'
etie - " --
Mx(t) = E(etX) =
i=o i!

The first two derivatives of Mx(t) are
M;(t) = AefeA(er- 1)
M;(t) = (Aet)2el(" - 1) + Aete"ec- 1)
Thus, by Eq. (4.42),
E(X) = Mi-0) = A E(X2) = M;(O) = A2 + A