Page 165 - Probability, Random Variables and Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 157

Let XI, . . . , Xn be n independent Cauchy r.v.'s with identical pdf shown in Prob. 4.58. Let

Find the characteristic function of Y, .
Find the pdf of Y, .
Does the central limit theorem hold?
From Eq. (4.1 24), the characteristic function of Xi is

Let Y = XI + . . + X,. Then the characteristic function of Y is

Now Y, = (l/n)Y. Thus, by Eq. (4.126), the characteristic function of Y, is

Equation (4.135) indicates that Y, is also a Cauchy r.v. with parameter a, and its pdf is the same as that
of Xi.
Since the characteristic function of Y, is independent of n and so is its pdf, Y, does not tend to a normal
r.v. as n + a, and so the central limit theorem does not hold in this case.

Let Y be a binomial r.v. with parameters (n, p). Using the central limit theorem, derive the
approximation formula

where Wz) is the cdf of a standard normal r.v. [Eq. (2.54)J.
We saw in Prob. 4.53 that if XI, . . . , X, are independent Bernoulli r.v.3, each with parameter p, then
-
Y = X, + . + X, is a binomial r.v. with parameters (n, p). Since X,'s are independent, we can apply the
central limit theorem to the r.v. 2, defined by

Thus, for large n, 2, is normally distributed and

Substituting Eq. (4.1 37) into Eq. (4.1 38) gives

Because we are approximating a discrete distribution by a continuous one, a slightly better approx-
imation is given by

Formula (4.139) is referred to as a continuity correction of Eq. (4.1 36).