Page 108 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES 101

Fig. 3-7
(c) The region in the xy plane corresponding to the event (0 < X < 3,O < Y < 3) is shown in Fig. 3-7 as
the shaded area R,. Then

Note that the bivariate r.v. (X, Y) is said to be uniformly distributed over the region Rxy if its pdf is
Y) = { k (x, Y) E RXY
fX&
0 otherwise
where k is a constant. Then by Eq. (3.26), the contant k must be k = l/(area of Rxy).

3.21. Suppose we select one point at random from within the circle with radius R. If we let the center
of the circle denote the origin and define X and Y to be the coordinates of the point chosen (Fig.
3-8), then (X, Y) is a uniform bivariate r.v. with joint pdf given by

where k is a constant.
(a) Determine the value of k.
(b) Find the marginal pdf's of X and Y.
(c) Find the probability that the distance from the origin of the point selected is not greater
than a.

Fig. 3-8