Page 166 - Probability, Random Variables and Random Processes

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

4.68. Let Y be a Poisson r.v. with parameter I. Using the central limit theorem, derive approximation
formula :

We saw in Prob. 4.54 that if XI, . . . , X, are independent Poisson r.v.'s Xi having parameter Ai, then
Y = XI + . . + X, is also a Poisson r.v. with parameter 1 = 1, + . . . + 1,. Using this fact, we can view a
Poisson r.v. Y with parameter I as a sum of independent Poisson r.v.'s Xi, i = 1, . . . , n, each with parameter
1/n; that is,

The central limit theorem then implies that the r.v. Z defined by

is approximately normal and
P(Z I z) w @(z)
Substituting Eq. (4.141) into Eq. (4.142) gives

P(? I z) = P(Y I Jiz + I) iE Wz)

Again, using a continuity correction, a slightly better approximation is given by

Supplementary Problems

4.69. Let Y = 2X + 3. Find the pdf of Y if X is a uniform r.v. over (- 1, 2).
Am. fy(y) = {i l<y<7
otherwise

4.70. Let X be a r.v. with pdf fx(x). Let Y = I X I. Find the pdf of Y in terms of f,(x).

Ans. fb)
=
Y <o
4.71. Let Y = sin X, where X is uniformly distributed over (0, 2x). Find the pdf of Y
I'
Ans. fro) rrJm -l<y<l
=
(0 otherwise