Page 77 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

(a) Find the probability that more than three calls will arrive during any 10-minute period.
(b) Find the probability that no calls will arrive during any 10-minute period.
(a) From Eq. (2.40), the pmf of X is

2k
Thus, P(X > 3) = -P(X I3) = 1 - e-2 -
k=O k!
= 1 - e-2(1 + 2 + 4 + 8) z 0.143
(b) P(X=0)=pdO)=e-2x0.135

243. Consider the experiment of throwing a pair of fair dice.
(a) Find the probability that it will take less than six tosses to throw a 7.
(b) Find the probability that it will take more than six tosses to throw a 7.
(a) From Prob. 1.31(a), we see that the probability of throwing a 7 on any toss is 4. Let X denote the
number of tosses required for the first success of throwing a 7. Then, from Prob. 2.15, it is clear that X
is a geometric r.v. with parameter p = 6. Thus, using Eq. (2.71) of Prob. 2.15, we obtain

(b) Similarly, we get
P(X > 6) = 1 - P(X 5 6) = 1 - FA6)
= 1 - [l - (2)6] = (:)6 w 0.335

2.44. Consider the experiment of rolling a fair die. Find the average number of rolls required in order
to obtain a 6.
Let X denote the number of trials (rolls) required until the number 6 first appears. Then X is a
geometrical r.v. with parameter p = 4. From Eq. (2.85) of Prob. 2.27, the mean of X is given by

Thus, the average number of rolls required in order to obtain a 6 is 6.

2.45. Assume that the length of a phone call in minutes is an exponential r.v. X with parameter
I = $. If someone arrives at a phone booth just before you arrive, find the probability that you
will have to wait (a) less than 5 minutes, and (b) between 5 and 10 minutes.
(a) From Eq. (2.48), the pdf of X is

Then

(b) Similarly,