Page 290 - Applied statistics and probability for engineers

268 Chapter 7/Point Estimation of Parameters and Sampling Distributions

(a) Find the likelihood function based on a random sample of 7-56. Suppose that X is a normal random variable with unknown
2
size n. Find the log likelihood. mean μ and known variance σ . The prior distribution for μ is a
(b) Show that the log likelihood is maximized by solving the uniform distribution deined over the interval [ , ]a b .
following equations (a) Find the posterior distribution for μ.
(b) Find the Bayes estimator for μ.
⎡ n β n ( ) ⎤ −1 7-57. Suppose that X is a Poisson random variable with
⎢ ∑ x i ln ( ) ∑ ln x i ⎥
x i
β = ⎢ = i 1 − = i 1 ⎥ parameter λ. Let the prior distribution for λ be a gamma distri-
(
⎢ n β n ⎥ bution with parameters m + 1 and m + ) λ1 / 0 .
⎢ ⎣ ∑ x i ⎥ ⎦ (a) Find the posterior distribution for λ.
= i 1
⎡ n β ⎤ 1/ β (b) Find the Bayes estimator for λ.
⎢ ∑ x i ⎥ 7-58. Suppose that X is a normal random variable with
=
δ = ⎢ i=1 ⎥ unknown mean and known variance σ = 9. The prior distribu-
2
2
⎢ n ⎥ tion for μ is normal with μ 0 = 4 and σ = 1. A random sample of
⎣ ⎦ 0
n = 25 observations is taken, and the sample mean is x = .4 85 .
(c) What complications are involved in solving the two equa- (a) Find the Bayes estimate of μ.
tions in part (b)? (b) Compare the Bayes estimate with the maximum likelihood

7-54. Reconsider the oxide thickness data in Exercise 7-35 estimate.
and suppose that it is reasonable to assume that oxide thickness 7-59. The weight of boxes of candy is a normal random
is normally distributed. variable with mean μ and variance 1 10/ pound. The prior dis-
2
(a) Compute the maximum likelihood estimates of μ and σ . tribution for μ is normal with mean 5.03 pound and variance
2
(b) Graph the likelihood function in the vicinity of ˆ μ and ˆ σ , the 1 25 pound. A random sample of 10 boxes gives a sample
/
maximum likelihood estimates, and comment on its shape. mean of x = .5 05 pounds.
(c) Suppose that the sample size was larger (n = 40 ) but the (a) Find the Bayes estimate of μ.
maximum likelihood estimates were numerically equal to (b) Compare the Bayes estimate with the maximum likelihood
the values obtained in part (a). Graph the likelihood function estimate.
for n = 40, compare it to the one from part (b), and comment 7-60. The time between failures of a machine has an expo-
on the effect of the larger sample size. nential distribution with parameter λ. Suppose that the prior distri-
7-55. Suppose that X is a normal random variable with unknown bution for λ is exponential with mean 100 hours. Two machines are
2
mean μ and known variance σ . The prior distribution for μ is a observed, and the average time between failures is x = 1125 hours.
2
normal distribution with mean μ 0 and variance σ 0 . Show that the (a) Find the Bayes estimate for λ.
Bayes estimator for μ becomes the maximum likelihood estimator (b) What proportion of the machines do you think will fail
when the sample size n is large. before 1000 hours?
Supplemental Exercises

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
7-61. Transistors have a life that is exponentially distrib- independently distributed with mean 105 ohms and standard
uted with parameter λ. A random sample of n transistors is taken. deviation of 2.0 ohms. What is the sampling distribution of
What is the joint probability density function of the sample? X 1 − X 2 ? What is the standard error of X 1 − X 2 ?
7-62. Suppose that a random variable is normally distributed 7-65. A random sample of 36 observations has been
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with mean μ and variance σ , and we draw a random sample of drawn from a normal distribution with mean 50 and stand-
ive observations from this distribution. What is the joint prob- ard deviation 12. Find the probability that the sample mean
ability density function of the sample? is in the interval 47 ≤ X ≤ 53. Is the assumption of normality
7-63. Suppose that X is uniformly distributed on the important? Why?
interval from 0 to 1. Consider a random sample of size 4 from 7-66. A random sample of n = 9 structural elements is
X. What is the joint probability density function of the sample? tested for compressive strength. We know that the true mean
7-64. A procurement specialist has purchased 25 resis- compressive strength μ = 5500 psi and the standard deviation
tors from vendor 1 and 30 resistors from vendor 2. Let is σ = 100 psi. Find the probability that the sample mean com-
X 11 , , X 1 2, ,..., X 1 25, represent the vendor 1 observed resistances, pressive strength exceeds 4985 psi.
which are assumed to be normally and independently dis- 7-67. A normal population has a known mean 50 and known
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tributed with mean 100 ohms and standard deviation 1.5 variance σ = 2. A random sample of n = 16 is selected from
ohms. Similarly, let X 2 1 , , X 2 2, ,..., X 2 30, represent the vendor 2 this population, and the sample mean is x = 52 . How unusual
observed resistances, which are assumed to be normally and is this result?