Page 154 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 147

Thus, solving the system (4.1 O3), we get
Xi = ~(YI +2Y2 + ~3)
x2 = 3 ~ -Y2 +Y3)
1
x3 = 4(Y, - 372 - 2~3)
Then by Eq. (4.31), we obtain

Since XI, X2 , and X3 are independent,

1
Hence, ~Y~Y~Y,(Y~, ~3)
=
~2,
where

EXPECTATION
436. Let X be a uniform r.v. over (0, 1) and Y = ex.
(a) Find E(Y) by using f,(y).
(b) Find E(Y) by using f,(x).
(a) From Eq. (4.76) (Prob. 4.9),

(0 otherwise

Hence,

(b) The pdf of X is
1 O<x<l
fx(x) = {O
otherwise
Then, by Eq. (4.33),

4.37. Let Y = ax + b, where a and b are constants. Show that

We verify for the continuous case. The proof for the discrete case is similar.