Page 84 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

Ans. (a) k = 1.
(b) Mixed r.v.
(c) (i) $; (ii) ; (iii) 0

2.67. It is known that the time (in hours) between consecutive traffic accidents can be described by the exponen-
tial r.v. X with parameter 1 = &. Find (i) P(X I 60); (ii) P(X > 120); and (iii) P(10 < X I 100).
Ans. (i) 0.632; (ii) 0.135; (iii) 0.658

2.68. Binary data are transmitted over a noisy communication channel in block of 16 binary digits. The probabil-
ity that a received digit is in error as a result of channel noise is 0.01. Assume that the errors occurring in
various digit positions within a block are independent.
(a) Find the mean and the variance of the number of errors per block.
(b) Find the probability that the number of errors per block is greater than or equal to 4.

Ans. (a) E(X) = 0.16, Var(X) = 0.1 58
(b) 0.165 x

2.69. Let the continuous r.v. X denote the weight (in pounds) of a package. The range of weight of packages is
between 45 and 60 pounds.

(a) Determine the probability that a package weighs more than 50 pounds.
(b) Find the mean and the variance of the weight of packages.
Hint: Assume that X is uniformly distributed over (45, 60).
Ans. (a) 4; (b) E(X) = 52.5, Var(X) = 18.75

2.70. In the manufacturing of computer memory chips, company A produces one defective chip for every nine
good chips. Let X be time to failure (in months) of chips. It is known that X is an exponential r.v. with
parameter 1 = f for a defective chip and A = with a good chip. Find the probability that a chip pur-
chased randomly will fail before (a) six months of use; and (b) one year of use.

Ans. (a) 0.501; (b) 0.729
2.71. The median of a continuous r.v. X is the value of x = x, such that P(X 2 x,) = P(X I The mode of X
x,).
is the value of x = x, at which the pdf of X achieves its maximum value.
(a) Find the median and mode of an exponential r.v. X with parameter 1.
(b) Find the median and mode of a normal r.v. X = N(p, a2).
Ans. (a) x, = (In 2)/1 = 0.69311, x, = 0
(b) x, = x, = p
2.72. Let the r.v. X denote the number of defective components in a random sample of n components, chosen
without replacement from a total of N components, r of which are defective. The r.v. X is known as the
hypergeometric r.v. with parameters (N, r, n).
(a) Find the prnf of X.
(b) Find the mean and variance of X.
Hint: To find E(X), note that
() = x ( x-1 ) and (:) = (L)(~
-
-
r,
x=O n-x
To find Var(X), first find E[X(X - I)].