Page 82 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

and hence by Eq. (2.1 13),

Thus, by Eq. (2.49), we conclude that X is an exponential r.v. with parameter 1 = fx(0) (>0).
Note that the memoryless property Eq. (2.1 10) is also known as the Markov property (see Chap. 5), and
it may be equivalently expressed as

Let X be the lifetime (in hours) of a component. Then Eq. (2.114) states that the probability that the
component will operate for at least x + t hours given that it has been operational for t hours is the same as
the initial probability that it will operate for at least x hours. In other words, the component "forgets" how
long it has been operating.
Note that Eq. (2.115) is satisfied when X is an exponential rev., since P(X > x) = 1 - FAX) = e-" and
e-A(x+t) = e-ki -At .
e

Supplementary Problems

2.54. Consider the experiment of tossing a coin. Heads appear about once out of every three tosses. If this
experiment is repeated, what is the probability of the event that heads appear exactly twice during the first
five tosses?
Ans. 0.329

2.55. Consider the experiment of tossing a fair coin three times (Prob. 1.1). Let X be the r.v. that counts the
number of heads in each sample point. Find the following probabilities:
(a) P(X I 1); (b) P(X > 1); and (c) P(0 < X < 3).

2.56. Consider the experiment of throwing two fair dice (Prob. 1.31). Let X be the r.v. indicating the sum of the
numbers that appear.
(a) What is the range of X?
(b) Find (i) P(X = 3); (ii) P(X 5 4); and (iii) P(3 < X 1 7).
Ans. (a) Rx = (2, 3,4, . . . , 12)
(b) (i) & ; (ii) 4 ; (iii) 4
2.57. Let X denote the number of heads obtained in the flipping of a fair coin twice.

(a) Find the pmf of X.
(b) Compute the mean and the variance of X.

2.58. Consider the discrete r.v. X that has the pmf
px(xk) = (JP xk = 1, 2, 3, . . .
Let A = (c: X({) = 1, 3, 5, 7, . . .}. Find P(A).
Ans. 3