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186 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

are 10, 20, and 70%, respectively. Assume that people react (c) Are the number of calls answered in two rings or less and
independently and that 20 people are given the medication. the number of calls requiring five rings or more independ-
Determine the following: ent random variables?
(a) The probability that 2, 4, and 14 people will suffer severe, 5-104. Determine the value of c such that the function
moderate, or minor side effects, respectively f(x, y) cx y for 0 x 3 and 0 y 2 satisfies the
2
(b) The probability that no one will suffer severe side effects properties of a joint probability density function.
(c) The mean and variance of the number of people that will
5-105. Continuation of Exercise 5-104. Determine the
suffer severe side effects
following:
(d) What is the conditional probability distribution of the
(a) P1X 1, Y 12 (b) P1X 2.52
number of people who suffer severe side effects given that
(c) P11 Y 2.52 (d) P1X
2, 1 Y 1.52
19 suffer minor side effects?
(e) E1X 2 (f) E1Y 2
(e) What is the conditional mean of the number of people who
5-106. Continuation of Exercise 5-104.
suffer severe side effects given that 19 suffer minor side
(a) Determine the marginal probability distribution of the
effects?
random variable X.
5-100. The backoff torque required to remove bolts in a
(b) Determine the conditional probability distribution of Y
steel plate is rated as high, moderate, or low. Historically, the
given that X 1.
probability of a high, moderate, or low rating is 0.6, 0.3, or
(c) Determine the conditional probability distribution of X
0.1, respectively. Suppose that 20 bolts are evaluated and that
given that Y 1.
the torque ratings are independent.
5-107. The joint distribution of the continuous random
(a) What is the probability that 12, 6, and 2 bolts are rated as
2
2
variables X, Y, and Z is constant over the region x y 1,
high, moderate, and low, respectively?
0 z 4.
(b) What is the marginal distribution of the number of bolts
2
2
(a) Determine P1X Y 0.52
rated low?
2
2
(b) Determine P1X Y 0.5, Z 22
(c) What is the expected number of bolts rated low?
(c) What is the joint conditional probability density function
(d) What is the probability that the number of bolts rated low
of X and Y given that Z 1?
is greater than two?
(d) What is the marginal probability density function of X?
5-101. Continuation of Exercise 5-100
5-108. Continuation of Exercise 5-107.
(a) What is the conditional distribution of the number of bolts
rated low given that 16 bolts are rated high? (a) Determine the conditional mean of Z given that X 0 and
(b) What is the conditional expected number of bolts rated Y 0.
low given that 16 bolts are rated high? (b) In general, determine the conditional mean of Z given that
(c) Are the numbers of bolts rated high and low independent X x and Y y.
random variables? 5-109. Suppose that X and Y are independent, continuous
5-102. To evaluate the technical support from a computer uniform random variables for 0 x 1 and 0 y 1. Use
manufacturer, the number of rings before a call is answered by the joint probability density function to determine the proba-
a service representative is tracked. Historically, 70% of the bility that 0 X Y 0 0.5.
calls are answered in two rings or less, 25% are answered in 5-110. The lifetimes of six major components in a copier are
three or four rings, and the remaining calls require five rings independent exponential random variables with means of 8000,
or more. Suppose you call this manufacturer 10 times and 10,000, 10,000, 20,000, 20,000, and 25,000 hours, respectively.
assume that the calls are independent. (a) What is the probability that the lifetimes of all the compo-
(a) What is the probability that eight calls are answered in two nents exceed 5000 hours?
rings or less, one call is answered in three or four rings, (b) What is the probability that at least one component life-
and one call requires five rings or more? time exceeds 25,000 hours?
(b) What is the probability that all 10 calls are answered in 5-111. Contamination problems in semiconductor manu-
four rings or less? facturing can result in a functional defect, a minor defect, or
(c) What is the expected number of calls answered in four no defect in the final product. Suppose that 20, 50, and 30% of
rings or less? the contamination problems result in functional, minor, and no
5-103. Continuation of Exercise 5-102 defects, respectively. Assume that the effects of 10 contamina-
(a) What is the conditional distribution of the number of calls tion problems are independent.
requiring five rings or more given that eight calls are (a) What is the probability that the 10 contamination problems
answered in two rings or less? result in two functional defects and five minor defects?
(b) What is the conditional expected number of calls requir- (b) What is the distribution of the number of contamination
ing five rings or more given that eight calls are answered problems that result in no defects?
in two rings or less?