Page 291 - Applied statistics and probability for engineers

Section 7-4/Methods of Point Estimation 269

θ
7-68. A random sample of size n = 16 is taken from a nor- 7-73. Let f x( ) (= 1 / ) x (1 −θ) / θ , 0 < x < , and 0 < <θ ∞. Show
1
2
ˆ
mal population with μ = 40 and σ = 5. Find the probability that Θ = −(1 / )Σn i n =1 ln( )X i is the maximum likelihood estimator
that the sample mean is less than or equal to 37. ˆ
for θ and that Θ is an unbiased estimator for q.
7-69. A manufacturer of semiconductor devices takes a ran-
7-74. You plan to use a rod to lay out a square, each side
dom sample of 100 chips and tests them, classifying each chip as
of which is the length of the rod. The length of the rod is μ,
defective or nondefective. Let X i = 0 if the chip is nondefective
which is unknown. You are interested in estimating the area
and X i = 1 if the chip is defective. The sample fraction defective is 2
of the square, which is μ . Because μ is unknown, you meas-
X + X +⋅⋅⋅+ ure it n times, obtaining observations X X 2 ,… , X n . Suppose
1 ,
P = 1 2 X 100 2
ˆ
100 that each measurement is unbiased for μ with variance σ .
2
ˆ
What is the sampling distribution of the random variable P? (a) Show that X is a biased estimate of the area of the square.
7-70. Let X be a random variable with mean (b) Suggest an estimator that is unbiased.
2
μ and variance σ . Given two independent random samples of 7-75. An electric utility has placed special meters on 10
sizes n 1 and n 2 , with sample means X 1 and X 2 , show that houses in a subdivision that measures the energy consumed
X = aX +(1 − ) 2 < a <1 (demand) at each hour of the day. The company is interested
a X , 0
in the energy demand at one speciic hour—the hour at which
1
the system experiences the peak consumption. The data from
is an unbiased estimator for μ. If X 1 and X 2 are independent,
these 10 meters are as follows (in KW): 23.1, 15.6, 17.4, 20.1,
ind the value of a that minimizes the standard error of X.
7-71. A random variable x has probability density function 19.8, 26.4, 25.1, 20.5, 21.9, and 28.7. If μ is the true mean peak
demand for the 10 houses in this group of houses having the
θ
∞
f x ( ) = 1 x e − x/ θ , 0 < x < , 0 < < ∞ special meters, estimate μ. Now suppose that the utility wants
2
2 θ 3
to estimate the demand at the peak hour for all 5000 houses in
Find the maximum likelihood estimator for θ. this subdivision. Let θ be this quantity. Estimate θ using the
7-72. Let f x ( ) = θ x θ−1 , 0 < < , and 0 < x < . Show that data given. Estimate the proportion of houses in the subdivision
θ
∞
1
ˆ n
= X i ) is the maximum likelihood estimator for θ.
ln
i
Θ = − n / ( Π 1 that demand at least 20 KW at the hour of system peak.
Mind-Expanding Exercises
7-76. A lot consists of N transistors, and of these, measured values for the ith part. Assume that these two ran-
M M ≤ N) are defective. We randomly select two transistors dom variables are independent and normally distributed and
(
2
without replacement from this lot and determine whether that both have true mean μ and variance σ .
i
2
they are defective or nondefective. The random variable (a) Show that the maximum likelihood estimator of σ is
2
2
( /
⎧ , 1 if the ith transistor ˆ σ = 1 4n )Σ n i =1 (X i −Y i ) .
⎪ (b) Show that σ is a biased estimator for ˆ σ . What happens
ˆ
2
2
⎪

X i = ⎨ is nondefective i = 2 , 1 to the bias as n becomes large?
s
⎪ , 0 if the ith transistor (c) Find an unbiased estimator for σ .
2
⎪ is defective 7-79. Consistent Estimator. Another way to measure the

⎩
and X 2 . ˆ
Determine the joint probability function for X 1 closeness of an estimator Θ to the parameter θ is in terms
ˆ
What are the marginal probability functions for X 1 and X 2 ? of consistency. If Θ n is an estimator of θ based on a random
ˆ
Are X 1 and X 2 independent random variables? sample of n observations, Θ n is consistent for θ if
7-77. When the sample standard deviation is based on a lim P( ⏐Θ − θ⏐
ˆ
e
random sample of size n from a normal population, it can n→∞ n < ) = 1
be shown that S is a biased estimator for σ. Speciically, Thus, consistency is a large-sample property describing the
ˆ
(
(
/ )
E S) = σ 2 / n −1 ) Γ n ( 2/ ) Γ ⎡ ⎣ n ( −1 2 ⎤ ⎦ limiting behavior of Θ n as n tends to ininity. It is usually
/
dificult to prove consistency using this deinition, although it
(a) Use this result to obtain an unbiased estimator for σ can be done from other approaches. To illustrate, show that X
of the form c S, when the constant c n depends on the is a consistent estimator of μ (when σ < ∞) by using Cheby-
2
n
sample size n. shev’s inequality from the supplemental material on the Web.
(b) Find the value of c for n = 10 and n = 25. Generally, 7-80. Order Statistics. Let X X 2 ,…
n 1 , , X n be a ran-
how well does S perform as an estimator of σ for large dom sample of size n from X , a random variable hav-
n with respect to bias? ing distribution function F x( ). Rank the elements in
7-78. An operator using a gauge measure collection of order of increasing numerical magnitude, resulting
n randomly selected parts twice. Let X i and Y i denote the in X ( ) , X ( ) ,… , X n ( ) , where X ( )1 is the smallest sample
1
2