Page 73 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

Thus, by Eq. (2.31), the variance of X is

2.33. Let X = N(p; a2). Verify Eqs. (2.57) and (2.58).
Using Eqs. (2.52) and (2.26), we have

Writing x as (x - p) + p, we have

Letting y = .x - p in the first integral, we obtain

The first integral is zero, since its integrand is an odd function. Thus, by the property of pdf Eq. (2.22), we
get

Next, by Eq. (2.29),

From Eqs. (2.22) and (2.52), we have

[~e-(x-~)2/~2az) dx =

Differentiating with respect to a, we obtain

Multiplying both sides by a2/&, we have

Thus, ax2 = Var(X) = a2

2.34. Find the mean and variance of a Rayleigh r.v. defined by Eq. (2.74) (Prob. 2.23).
Using Eqs. (2.74) and (2.26), we have

Now the variance of N(0; a2) is given by