Page 211 - Applied Statistics And Probability For Engineers

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5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES 187

(c) What is the expected number of contamination problems (a) What is the probability that the weight of an assembly
that result in no defects? exceeds 29.5 ounces?
5-112. The weight of adobe bricks for construction is (b) What is the probability that the mean weight of eight
normally distributed with a mean of 3 pounds and a standard independent assemblies exceeds 29 ounces?
deviation of 0.25 pound. Assume that the weights of the bricks 5-117. Suppose X and Y have a bivariate normal distribution
are independent and that a random sample of 25 bricks is with X 4 , Y 1 , X 4 , Y 4 , and 0.2 . Draw
chosen. a rough contour plot of the joint probability density function.
(a) What is the probability that the mean weight of the sample 1 1
is less than 2.95 pounds? 5-118. If f XY 1x, y2 exp e 31x 12 2
1.2 0.72
(b) What value will the mean weight exceed with probability
0.99? 2
1.61x 121y 22 1y 22 4f
5-113. The length and width of panels used for interior doors
determine E(X), E(Y), V(X), V(Y), and by recorganizing the
(in inches) are denoted as X and Y, respectively. Suppose that X
parameters in the joint probability density function.
and Y are independent, continuous uniform random variables for
17.75 x 18.25 and 4.75 y 5.25, respectively. 5-119. The permeability of a membrane used as a moisture
(a) By integrating the joint probability density function over barrier in a biological application depends on the thickness of
the appropriate region, determine the probability that the two integrated layers. The layers are normally distributed with
area of a panel exceeds 90 squared inches. means of 0.5 and 1 millimeters, respectively. The standard
(b) What is the probability that the perimeter of a panel deviations of layer thickness are 0.1 and 0.2 millimeters,
exceeds 46 inches? respectively. The correlation between layers is 0.7.
(a) Determine the mean and variance of the total thickness of
5-114. The weight of a small candy is normally distributed
the two layers.
with a mean of 0.1 ounce and a standard deviation of 0.01
(b) What is the probability that the total thickness is less than
ounce. Suppose that 16 candies are placed in a package and
1 millimeter?
that the weights are independent.
(c) Let X 1 and X 2 denote the thickness of layers 1 and 2, re-
(a) What are the mean and variance of package net weight?
spectively. A measure of performance of the membrane is
(b) What is the probability that the net weight of a package is
a function 2X 1 3X 2 of the thickness. Determine the
less than 1.6 ounces?
mean and variance of this performance measure.
(c) If 17 candies are placed in each package, what is the
probability that the net weight of a package is less than 5-120. The permeability of a membrane used as a moisture
1.6 ounces? barrier in a biological application depends on the thickness of
three integrated layers. Layers 1, 2, and 3 are normally dis-
5-115. The time for an automated system in a warehouse to
tributed with means of 0.5, 1, and 1.5 millimeters, respec-
locate a part is normally distributed with a mean of 45 seconds
tively. The standard deviations of layer thickness are 0.1, 0.2,
and a standard deviation of 30 seconds. Suppose that inde-
and 0.3, respectively. Also, the correlation between layers 1
pendent requests are made for 10 parts.
and 2 is 0.7, between layers 2 and 3 is 0.5, and between layers
(a) What is the probability that the average time to locate 10
1 and 3 is 0.3.
parts exceeds 60 seconds?
(a) Determine the mean and variance of the total thickness of
(b) What is the probability that the total time to locate 10
the three layers.
parts exceeds 600 seconds?
(b) What is the probability that the total thickness is less than
5-116. A mechanical assembly used in an automobile en-
1.5 millimeters?
gine contains four major components. The weights of the
5-121. A small company is to decide what investments to
components are independent and normally distributed with
use for cash generated from operations. Each investment has a
the following means and standard deviations (in ounces):
mean and standard deviation associated with the percentage
gain. The first security has a mean percentage gain of 5% with
a standard deviation of 2%, and the second security provides
Standard the same mean of 5% with a standard deviation of 4%. The
Component Mean Deviation securities have a correlation of 0.5, so there is a negative
correlation between the percentage returns. If the company
Left case 4 0.4
Right case 5.5 0.5 invests two million dollars with half in each security, what is
Bearing assembly 10 0.2 the mean and standard deviation of the percentage return?
Bolt assembly 8 0.5 Compare the standard deviation of this strategy to one that
invests the two million dollars into the first security only.