Page 85 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

2.73. A lot consisting of 100 fuses is inspected by the following procedure: Five fuses are selected randomly, and
if all five "blow" at the specified amperage, the lot is accepted. Suppose that the lot contains 10 defective
fuses. Find the probability of accepting the lot.
Hint: Let X be a r.v. equal to the number of defective fuses in the sample of 5 and use the result of Prob.
2.72.
Ans. 0.584

2.74. Consider the experiment of observing a sequence of Bernoulli trials until exactly r successes occur. Let the
r.v. X denote the number of trials needed to observe the rth success. The r-v. X is known as the negative
binomial r.v. with parameter p, where p is the probability of a success at each trial.

(a) Find the pmf of X.
(b) Find the mean and variance of X.
Hint: To find E(X), use Maclaurin's series expansions of the negative binomial h(q) = (1 - 9)-' and its
derivatives h'(q) and hW(q), and note that

To find Var(X), first find E[(X - r)(X - r - 1)] using hU(q).

Ans. (a) px(x) = x = r, r + 1, ...
r(l - P)
(b) EIX) = r(i), Var(X) = -
p2
2.75. Suppose the probability that a bit transmitted through a digital communication channel and received in
error is 0.1. Assuming that the transmissions are independent events, find the probability that the third
error occurs at the 10th bit.
Ans. 0.017

2.76. A r.v. X is called a Laplace r.v. if its pdf is given by
fx(x)=ke-'Ix1 1>0, -co<x<oo
where k is a constant.
(a) Find the value of k.
(b) Find the cdf of X.
(c) Find the mean and the variance of X.

2.77. A r.v. X is called a Cauchy r.v. if its pdf is given by