Page 144 - Probability, Random Variables and Random Processes

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

4.14. Let X be a continuous r.v. with the pdf

Find the transformation Y = g(X) such that the pdf of Y is

O<y<l

otherwise
The cdf of X is

Then from the result of Prob. 4.12, the r.v. Z = 1 - e-' is uniformly distributed over (0, 1). Similarly, the
cdf of Y is

otherwise

and the r.v. W = fi is uniformly distributed over (0, 1). Thus, by setting Z = W, the required transfor-
mation is Y = (1 - e-X)2.

FUNCTIONS OF TWO RANDOM VARIABLES
4.15. Consider Z = X + Y. Show that if X and Y are independent Poisson r.v.'s with parameters A,
and A,, respectively, then Z is also a Poisson r.v. with parameter A, + A,.
We can write the event

where events (X = i, Y = n - i), i = 0, 1, . . . , n, are disjoint. Since X and Y are independent, by Eqs. (1.46)
and (2.40), we have

n in - i n ~i~n-i
- Ali 2 - E L
- e-(h+h2'
- C e-h - ---A2
i=o i! (n - i)! i=o z! (n - i)!
e-(ai+n~) " n!
=- C- iin-i
n! i=,, t!(n - i)!

which indicates that Z = X + Y is a Poisson r.v. with A, + 1, .

4.16. Consider two r.v.'s X and Y with joint pdf f,,(x, y). Let Z = X + Y.
(a) Determine the pdf of Z.
(b) Determine the pdf of Z if X and Y are independent.