Page 218 - Applied statistics and probability for engineers

196 Chapter 5/Joint Probability Distributions

(b) What is the probability that all 10 calls are answered in four (c) What is the expected number of contamination problems
rings or less? that result in no defects?
(c) What is the expected number of calls answered in four 5-110. The weight of adobe bricks for construction is
rings or less? normally distributed with a mean of 3 pounds and a stand-
(d) What is the conditional distribution of the number of ard deviation of 0.25 pound. Assume that the weights of the
calls requiring ive rings or more given that eight calls are bricks are independent and that a random sample of 25 bricks
answered in two rings or less? is chosen.
(e) What is the conditional expected number of calls requiring (a) What is the probability that the mean weight of the sample
ive rings or more given that eight calls are answered in two is less than 2.95 pounds?
rings or less? (b) What value will the mean weight exceed with probability
(f) Is the number of calls answered in two rings or less and the 0.99?
number of calls requiring ive rings or more independent 5-111. The length and width of panels used for interior
random variables? doors (in inches) are denoted as X and Y, respectively. Sup-
5-105. Determine the value of c such that the function f(x, pose that X and Y are independent, continuous uniform ran-
2
y) = cx y for 0 , x , 3 and 0 , y , 2 satisies the properties of dom variables for 17.75 , x , 18.25 and 4.75 , y , 5.25,
a joint probability density function. respectively.
Determine the following: (a) By integrating the joint probability density function over
(
(
.
(a) P X < ,Y <1 1) (b) P X < 2 5) the appropriate region, determine the probability that the
(c) P <Y < 2 5) (d) P X > 2 1 <Y <1 5) area of a panel exceeds 90 square inches.
1 (
(
.
.
.
(e) E X ( ) (f) E Y ( ) (b) What is the probability that the perimeter of a panel exceeds
46 inches?
(g) Marginal probability distribution of the random variable X 5-112. The weight of a small candy is normally distrib-
(h) Conditional probability distribution of Y given that X = 1 uted with a mean of 0.1 ounce and a standard deviation of 0.01
(i) Conditional probability distribution of X given that Y = 1 ounce. Suppose that 16 candies are placed in a package and that
5-106. The joint distribution of the continuous random varia-
bles X, Y, and Z is constant over the region x + y ≤ 1 , < z < 4 . the weights are independent.
2
2
0
Determine the following: (a) What are the mean and variance of the package's net weight?
(
(
(b) What is the probability that the net weight of a package is
.
.
(a) P X + Y ≤ 0 5) (b) P X + Y ≤ 0 5 ,Z < 2) less than 1.6 ounces?
2
2
2
2
(c) Joint conditional probability density function of X and Y (c) If 17 candies are placed in each package, what is the prob-
given that Z = 1 ability that the net weight of a package is less than 1.6
(d) Marginal probability density function of X ounces?
(e) Conditional mean of Z given that X = 0 and Y = 0 5-113. The time for an automated system in a warehouse
(f) Conditional mean of Z given that X = x and Y = y to locate a part is normally distributed with a mean of 45 sec-
5-107. Suppose that X and Y are independent, continuous onds and a standard deviation of 30 seconds. Suppose that
uniform random variables for 0 , x , 1 and 0 , y , 1. Use the independent requests are made for 10 parts.
joint probability density function to determine the probability (a) What is the probability that the average time to locate 10
that ⏐X − Y ⏐< . . parts exceeds 60 seconds?
5
0
5-108. The lifetimes of six major components in a cop- (b) What is the probability that the total time to locate 10 parts
ier are independent exponential random variables with means exceeds 600 seconds?
of 8000, 10,000, 10,000, 20,000, 20,000, and 25,000 hours, 5-114. A mechanical assembly used in an automobile
respectively. engine contains four major components. The weights of the com-
(a) What is the probability that the lifetimes of all the compo- ponents are independent and normally distributed with the follow-
nents exceed 5000 hours? ing means and standard deviations (in ounces):
(b) What is the probability that at least one component's life-
time exceeds 25,000 hours? Component Mean Standard Deviation
5-109. Contamination problems in semiconductor manu-
facturing can result in a functional defect, a minor defect, or no Left case 4.0 0.4
defect in the inal product. Suppose that 20%, 50%, and 30% of Right case 5.5 0.5
the contamination problems result in functional, minor, and no Bearing assembly 10.0 0.2
defects, respectively. Assume that the defects of 10 contamina- Bolt assembly 8.0 0.5
tion problems are independent.
(a) What is the probability that the 10 contamination problems (a) What is the probability that the weight of an assembly
result in two functional defects and ive minor defects? exceeds 29.5 ounces?
(b) What is the distribution of the number of contamination (b) What is the probability that the mean weight of eight inde-
problems that result in no defects? pendent assemblies exceeds 29 ounces?