Page 67 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

Iff (x) is a pdf of a continuous r.v. X, then by Eq. (2.22), we must have

1
Now by Eq. (2.52), the pdf of N(i; 4) is - e-(x-'/2)2. Thus,
J;;

from which we obtain a = a.

2.23. A r.v. X is called a Rayleigh r.v. if its pdf is given by
#

(a) Determine the corresponding cdf FX(x).
(b) Sketch.fx(x) and FX(x) for a = 1.

(a) By Eq. (2.24), the cdf of X is

Let y = t2/(2a2). Then dy = (l/a2)t dt, and

(b) With a = 1, we have

and

These functions are sketched in Fig. 2-19.

2.24. A r.v. X is called a gamma r.v. with parameter (a, A) (a > 0 and 1 > 0) its pdf is given by
if

where T(a) is the gamma function defined by