Page 167 - Probability, Random Variables and Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 159

4.72. Let X and Y be independent r.v.3, each uniformly distributed over (0, 1). Let Z = X + Y, W = X - Y.
Find the marginal pdf s of Z and W.
O<z<l w+l -l<w<O
1 <z<2 f&w)= O<w<l
otherwise otherwise

4.73. Let X and Y be independent exponential r.v.'s with parameters a and B, respectively. Find the pdf of
(a) Z = X - Y; (b) Z = X/Y; (c) Z = max(X, Y); (d) Z = min(X, Y).

4.74. Let X denote the number of heads obtained when three independent tossings of a fair coin are made. Let
Y = X2. Find E(Y).
Ans. 3

4.75. Let X be a uniform rev. over (- 1, 1). Let Y = Xn.
(a) Calculate the covariance of X and Y.
(b) Calculate the correlation coefficient of X and Y

n = even n = even

4.76. Let the moment generating function of a discrete r.v. X be given by
Mx(t) = 0.25et + 0.35e3' + 0.40e5'
Find P(X = 3).
Ans. 0.35

4.77. Let X be a geometric r.v. with parameter p.
(a) Determine the moment generating function of X.
(b) Find the mean of X for p = 3.

pet
Ans. Mx(t) = -
1 - qe'
4.78. Let X be a uniform r.v. over (a, b).
(a) Determine the moment generating function of X.
(b) Using the result of (a), find E(X), E(X2), and E(X3).
erb -
Ans. (a) Mx(t) = ------
t(b - a)
(b) E(X) = $(b + a), E(X2) = 4(b2 + ab + a2) E(X3) = $(b3 + b2a + ba2 + a3)