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5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES 185

B C (b) Determine the number of wafers that needs to be meas-
ured such that the probability that the average thickness
D
exceeds 11 micrometers is 0.01.
B C
(c) If the mean thickness is 10 micrometers, what should the
standard deviation of thickness equal so that the probabil-
ity that the average of 10 wafers is either greater than 11 or
A less than 9 micrometers is 0.001?
5-95. Assume that the weights of individuals are independ-
A ent and normally distributed with a mean of 160 pounds and a
Figure 5-20 Figure for the standard deviation of 30 pounds. Suppose that 25 people
U-shaped component. squeeze into an elevator that is designed to hold 4300 pounds.
(a) What is the probability that the load (total weight) exceeds
the design limit?
(b) What design limit is exceeded by 25 occupants with prob-
(b) What is the probability that the width of the casing minus ability 0.0001?
the width of the door exceeds 1 4 inch?
(c) What is the probability that the door does not fit in the
casing? 5-8 FUNCTIONS OF RANDOM
5-92. A U-shaped component is to be formed from the three VARIABLES (CD ONLY)
parts A, B, and C. The picture is shown in Fig. 5-20. The length
of A is normally distributed with a mean of 10 millimeters and 5-9 MOMENT GENERATING
a standard deviation of 0.1 millimeter. The thickness of parts B FUNCTION (CD ONLY)
and C is normally distributed with a mean of 2 millimeters and
a standard deviation of 0.05 millimeter. Assume all dimensions
are independent. 5-10 CHEBYSHEV’S INEQUALITY
(a) Determine the mean and standard deviation of the length (CD ONLY)
of the gap D.
(b) What is the probability that the gap D is less than 5.9 mil- Supplemental Exercises
limeters?
5-93. Soft-drink cans are filled by an automated filling ma- 5-96. Show that the following function satisfies the proper-
chine and the standard deviation is 0.5 fluid ounce. Assume ties of a joint probability mass function:
that the fill volumes of the cans are independent, normal ran-
dom variables.
x y f(x, y)
(a) What is the standard deviation of the average fill volume
0 0 1 4
of 100 cans?
(b) If the mean fill volume is 12.1 ounces, what is the proba- 0 1 1 8
bility that the average fill volume of the 100 cans is below 1 0 1 8
12 fluid ounces? 1 1 1 4
(c) What should the mean fill volume equal so that the proba-
2 2 1 4
bility that the average of 100 cans is below 12 fluid ounces
is 0.005?
5-97. Continuation of Exercise 5-96. Determine the follow-
(d) If the mean fill volume is 12.1 fluid ounces, what should
ing probabilities:
the standard deviation of fill volume equal so that the
(a) P1X 0.5, Y 1.52 (b) P1X 12
probability that the average of 100 cans is below 12 fluid
ounces is 0.005? (c) P1X 1.52 (d) P1X
0.5, Y 1.52
(e) Determine the number of cans that need to be measured (e) Determine E(X), E(Y), V(X), and V(Y).
such that the probability that the average fill volume is 5-98. Continuation of Exercise 5-96. Determine the following:
less than 12 fluid ounces is 0.01. (a) Marginal probability distribution of the random variable X
5-94. The photoresist thickness in semiconductor manufac- (b) Conditional probability distribution of Y given that X 1
turing has a mean of 10 micrometers and a standard deviation of (c) E1Y 0 X 12
1 micrometer. Assume that the thickness is normally distributed (d) Are X and Y independent? Why or why not?
and that the thicknesses of different wafers are independent. (e) Calculate the correlation between X and Y.
(a) Determine the probability that the average thickness of 10 5-99. The percentage of people given an antirheumatoid
wafers is either greater than 11 or less than 9 micrometers. medication who suffer severe, moderate, or minor side effects