Page 110 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES 103

3.23. A manufacturer has been using two different manufacturing processes to make computer
memory chips. Let (X, Y) be a bivariate r.v., where X denotes the time to failure of chips made
by process A and Y denotes the time to failure of chips rnade by process B. Assuming that the
joint pdf of (X, Y) is

where a = and b = l.2(10-4), determine P(X > Y).

The region in the xy plane corresponding to the event (X > Y) is shown in Fig. 3-9 as the shaded area.
Then

Fig. 3-9

3.24. A smooth-surface table is ruled with equidistant parallel lines a distance D apart. A needle of
length L, where L I D, is randomly dropped onto this table. What is the probability that the
needle will intersect one of the lines? (This is known as BufJbn's needle problem.)
We can determine the needle's position by specifying a bivariate r.v. (X, 0), where X is the distance
from the middle point of the needle to the nearest parallel line: and O is the angle from the vertical to the
needle (Fig. 3-10). We interpret the statement "the needle is randomly dropped to mean that both X and O
have uniform distributions and that X and O are independent. The possible values of X are between 0 and

- -- - - - - - - - - -
Fig. 3-10 Buffon's needle problem.