Page 97 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

Let x be an n-dimensional vector (n x 1 matrix) defined by

X = [;I]
xn
The n-variate r.v. (XI, . . . , Xn) is called an n-variate normal r.v. if its joint pdf is given by

where T denotes the "transpose," p is the vector mean, K is the covariance matrix given by

and det K is the determinant of the matrix K. Note that f,(x) stands for f,, ... L(xl, . . . , xn).

Solved Problems

BIVARIATE RANDOM VARIABLES AND JOINT DISTRIBUTION FUNCTIONS
3.1. Consider an experiment of tossing a fair coin twice. Let (X, Y) be a bivariate r.v., where X is the
number of heads that occurs in the two tosses and Y is the number of tails that occurs in the two
tosses.
(a) What is the range Rx of X?
(b) What is the range Ry of Y?
(c) Find and sketch the range Rxy of (X, Y).
(d) Find P(X = 2, Y = 0), P(X = 0, Y = 2), and P(X = 1, Y = 1).
The sample space S of the experiment is
S = {HH, HT, TH, TT)

(a) R, = (0, 1,2)
(b) R, = (0, 192)
(c) RXY = ((2, O), (1, I), (0, 2)) which is sketched in Fig. 3-2.
(d) Since the coin is fair, we have
P(X = 2, Y = 0) = P(HH} = $
P(X = 0, Y = 2) = P{TT) = 4
P(X= 1, Y = 1)= P{HT, TH} =

3.2. Consider a bivariate r.v. (X, Y). Find the region of the xy plane corresponding to the events
A={X+Y12) B = {x2 + Y2 < 4)
C = {min(X, Y) 1 2) D = (max(X, Y) 1 2)