Page 164 - Probability, Random Variables and Random Processes

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

Let XI, .. ., X,, be a random sample of X with mean p and variance a2. How many
samples of X should be taken if the probability that the sample mean will not deviate from the
true mean p by more than a/10 is at least 0.95?
Setting 8 = u/10 in Eq. (4.1 Z9), we have

Thus if we want this probability to be at least 0.95, we must have 100/n < 0.05 or n 2 100/0.05 = 2000.

4.65. Verify the central limit theorem (4.61 ).
Let XI, ..., X, be a sequence of independent, identically distributed r.v.'s with E(Xi) = p and
Var(Xi) = a2. Consider the sum S, = X1 + . + X,. Then by Eqs. (4.108) and (4.112), we have E(S,) = np
and Var(S,) = na2. Let

Then by Eqs. (4.105) and (4.106), we have E(Zn) = 0 and Var(Z,) = 1. Let M(t) be the moment generating
function of the standardized r.v. = (Xi - ,u)/a. Since E(5) = 0 and E(X2) = Var(q) = 1, by Eq. (4.42), we
have

Given that Mf(t) and M"(t) are continuous functions of t, a Taylor (or Maclaurin) expansion of M(t) about
t = 0 can be expressed as

By adding and subtracting t2/2, we have

Now, by Eqs. (4.1 20) and (4.1 22), the moment generating function of 2, is

Using Eq. (4.1 32), Eq. (4.133) can be written as

where now t, is between 0 and t/&. Since M"(t) is continuous at t = 0 and t, -+ 0 as n + co, we have
lim [MU(t,) - 11 = M"(0) - 1 = 1 - 1 = 0
n+co
Thus, from elementary calculus, limn,, (1 + xln)" = ex, and we obtain

The right-hand side is the moment generating function of the standard normal r.v. Z = N(0; 1) [Eq.
(4.1 1911. Hence, by Lemma 4.2 of the moment generating function,
lim Z, = N(0; 1)
n+co