Page 71 - Schaum's Outlines - Probability, Random Variables And Random Processes

RANDOM VARIABLES [CHAP 2

and by Eq. (2.31),

2.30. Find the mean and variance of the r.v. X of Prob. 2.20.
From Prob. 2.20, the pdf of X is
2x O<x<l
fx(x) = (0
otherwise
By Eq. (2.26), the mean of X is

By Eq. (2.27), we have

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

2.31. Let X be a uniform r.v. over (a, b). Verify Eqs. (2.46) and (2.47).
By Eqs. (2.44) and (2.26), the mean of X is

By Eq. (2.27), we have

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

2.32. Let X be an exponential r.v. X with parameter A. Verify Eqs. (2.50) and (2.51).
By Eqs. (2.48) and (2.26), the mean of X is

Integrating by parts (u = x, du = Re-" dx) yields

Next, by Eq. (2.27),

Again integrating by parts (u = x2, du = le-" dx), we obtain