Page 210 - Applied statistics and probability for engineers

188 Chapter 5/Joint Probability Distributions

(e) Determine the number of cans that need to be measured that the covariance between X 1 and X 2 is 0.02. Determine the
such that the probability that the average ill volume is less variance of the perimeter Y = 2 X + 2 X 2 of a part. Compare and
1
than 12 oz is 0.01. comment on the result here and in the example.
5-71. The photoresist thickness in semiconductor manufac- 5-76. Three electron emitters produce electron beams with
turing has a mean of 10 micrometers and a standard deviation of changing kinetic energies that are uniformly distributed in the
,
1 micrometer. Assume that the thickness is normally distributed ranges [3 7, ], [2 5 and [4 10 Let Y denote the total kinetic
,
].
],
and that the thicknesses of different wafers are independent. energy produced by these electron emitters.
(a) Determine the probability that the average thickness of 10 (a) Suppose that the three beam energies are independent.
wafers is either greater than 11 or less than 9 micrometers. Determine the mean and variance of Y .
(b) Determine the number of wafers that need to be measured (b) Suppose that the covariance between any two beam ener-
such that the probability that the average thickness exceeds gies is − .0 5. Determine the mean and variance of Y .
11 micrometers is 0.01. (c) Compare and comment on the results in parts (a) and (b).
(c) If the mean thickness is 10 micrometers, what should the 5-77. In Exercise 5-31, the monthly demand for MMR vac-
standard deviation of thickness equal so that the probability cine was assumed to be approximately normally distributed
that the average of 10 wafers is either more than 11 or less with a mean and standard deviation of 1.1 and 0.3 million
than 9 micrometers is 0.001? doses, respectively. Suppose that the demands for different
5-72. Assume that the weights of individuals are inde- months are independent, and let Z denote the demand for a
pendent and normally distributed with a mean of 160 pounds year (in millions of does). Determine the following:
and a standard deviation of 30 pounds. Suppose that 25 people (a) Mean, variance, and distribution of Z
squeeze into an elevator that is designed to hold 4300 pounds. (b) P Z( <13 .2 )
<
(a) What is the probability that the load (total weight) exceeds (c) P(11< Z 15 )
the design limit? (d) Value for c such that P Z( < c) = .99
0
(b) What design limit is exceeded by 25 occupants with prob- 5-78. The rate of return of an asset is the change in price divided
ability 0.0001? by the initial price (denoted as r). Suppose that $10,000 is used
5-73. Weights of parts are normally distributed with variance to purchase shares in three stocks with rates of returns X X X 3 .
2 ,
1 ,
2
σ . Measurement error is normally distributed with mean 0 and Initially, $2500, $3000, and $4500 are allocated to each one,
2
variance 0.5σ , independent of the part weights, and adds to the respectively. After one year, the distribution of the rate of return
part weight. Upper and lower speciications are centered at 3σ for each is normally distributed with the following parameters:
.
.
about the process mean. μ 1 = 0 12, σ 1 = 0 14, μ 2 = 0 04, σ 2 = 0 02, μ 3 = 0 07, σ 3 = 0 08.
.
.
.
.
(a) Without measurement error, what is the probability that a (a) Assume that these rates of return are independent. Determine
part exceeds the speciications? the mean and variance of the rate of return after one year for
(b) With measurement error, what is the probability that a part the entire investment of $10,000.
is measured as being beyond speciications? Does this (b) Assume that X 1 is independent of X 2 and X 3 but that the
imply it is truly beyond speciications? covariance between X 2 and X 3 is − .0 005. Repeat part (a).
(c) What is the probability that a part is measured as being (c) Compare the means and variances obtained in parts (a) and
beyond speciications if the true weight of the part is 1 σ (b) and comment on any beneits from negative covari-
below the upper speciication limit? ances between the assets.
5-74. A U-shaped component is to be formed from the three
parts A, B, and C. See Fig. 5-18. The length of A is normally B C
distributed with a mean of 10 mm and a standard deviation of 0.1 D
mm. The thickness of parts B and C is normally distributed with B C
a mean of 2 mm and a standard deviation of 0.05 mm. Assume
that all dimensions are independent.
(a) Determine the mean and standard deviation of the length
of the gap D. A
(b) What is the probability that the gap D is less than 5.9 mm?
5-75. Consider the perimeter of a part in Example 5-32. Let A
denote the length and width of a part with stand-
X 1 and X 2
ard deviations 0.1 and 0.2 centimeters, respectively. Suppose FIGURE 5-18 Illustration for the U-shaped component.

5-5 General Functions of Random Variables

In many situations in statistics, it is necessary to derive the probability distribution of a func-
tion of one or more random variables. In this section, we present some results that are helpful
in solving this problem.