186 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONSare 10, 20, and 70%, respectively. Assume that people react
independently and that 20 people are given the medication.
Determine the following:
(a) The probability that 2, 4, and 14 people will suffer severe,
moderate, or minor side effects, respectively
(b) The probability that no one will suffer severe side effects
(c) The mean and variance of the number of people that will
suffer severe side effects
(d) What is the conditional probability distribution of the
number of people who suffer severe side effects given that
19 suffer minor side effects?
(e) What is the conditional mean of the number of people who
suffer severe side effects given that 19 suffer minor side
effects?
5-100. The backoff torque required to remove bolts in a
steel plate is rated as high, moderate, or low. Historically, the
probability of a high, moderate, or low rating is 0.6, 0.3, or
0.1, respectively. Suppose that 20 bolts are evaluated and that
the torque ratings are independent.
(a) What is the probability that 12, 6, and 2 bolts are rated as
high, moderate, and low, respectively?
(b) What is the marginal distribution of the number of bolts
rated low?
(c) What is the expected number of bolts rated low?
(d) What is the probability that the number of bolts rated low
is greater than two?
5-101. Continuation of Exercise 5-100
(a) What is the conditional distribution of the number of bolts
rated low given that 16 bolts are rated high?
(b) What is the conditional expected number of bolts rated
low given that 16 bolts are rated high?
(c) Are the numbers of bolts rated high and low independent
random variables?
5-102. To evaluate the technical support from a computer
manufacturer, the number of rings before a call is answered by
a service representative is tracked. Historically, 70% of the
calls are answered in two rings or less, 25% are answered in
three or four rings, and the remaining calls require five rings
or more. Suppose you call this manufacturer 10 times and
assume that the calls are independent.
(a) What is the probability that eight calls are answered in two
rings or less, one call is answered in three or four rings,
and one call requires five rings or more?
(b) What is the probability that all 10 calls are answered in
four rings or less?
(c) What is the expected number of calls answered in four
rings or less?
5-103. Continuation of Exercise 5-102
(a) What is the conditional distribution of the number of calls
requiring five rings or more given that eight calls are
answered in two rings or less?
(b) What is the conditional expected number of calls requir-
ing five rings or more given that eight calls are answered
in two rings or less?(c) Are the number of calls answered in two rings or less and
the number of calls requiring five rings or more independ-
ent random variables?
5-104. Determine the value of csuch that the function
f(x,y)cx^2 yfor 0x3 and 0y2 satisfies the
properties of a joint probability density function.
5-105. Continuation of Exercise 5-104. Determine the
following:
(a) (b)
(c) (d)
(e) (f)
5-106. Continuation of Exercise 5-104.
(a) Determine the marginal probability distribution of the
random variable X.
(b) Determine the conditional probability distribution of Y
given that X1.
(c) Determine the conditional probability distribution of X
given that Y1.
5-107. The joint distribution of the continuous random
variables X, Y, and Zis constant over the region(a) Determine
(b) Determine
(c) What is the joint conditional probability density function
of Xand Ygiven that Z1?
(d) What is the marginal probability density function of X?
5-108. Continuation of Exercise 5-107.
(a) Determine the conditional mean of Zgiven that X0 and
Y0.
(b) In general, determine the conditional mean of Zgiven that
Xxand Yy.
5-109. Suppose that Xand Yare independent, continuous
uniform random variables for 0x1 and 0y1. Use
the joint probability density function to determine the proba-
bility that
5-110. The lifetimes of six major components in a copier are
independent exponential random variables with means of 8000,
10,000, 10,000, 20,000, 20,000, and 25,000 hours, respectively.
(a) What is the probability that the lifetimes of all the compo-
nents exceed 5000 hours?
(b) What is the probability that at least one component life-
time exceeds 25,000 hours?
5-111. Contamination problems in semiconductor manu-
facturing can result in a functional defect, a minor defect, or
no defect in the final product. Suppose that 20, 50, and 30% of
the contamination problems result in functional, minor, and no
defects, respectively. Assume that the effects of 10 contamina-
tion problems are independent.
(a) What is the probability that the 10 contamination problems
result in two functional defects and five minor defects?
(b) What is the distribution of the number of contamination
problems that result in no defects?(^0) XY (^0) 0.5.
P 1 X^2 Y^2 0.5, Z 22
P 1 X^2 Y^2 0.5 2
0 z4.
x^2 y^2 1,
E 1 X 2 E 1 Y 2
P 11 Y2.5 2 P 1 X 2, 1Y1.5 2
P 1 X1, Y 12 P 1 X2.5 2
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