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7-5 SAMPLING DISTRIBUTIONS OF MEANS 245

MIND-EXPANDING EXERCISES

7-61. A lot consists of N transistors, and of these M Thus, consistency is a large-sample property, describing
ˆ
(M N) are defective. We randomly select two transis- the limiting behavior of
n as n tends to infinity. It is
tors without replacement from this lot and determine usually difficult to prove consistency using the above
whether they are defective or nondefective. The ran- definition, although it can be done from other ap-
dom variable proaches. To illustrate, show that X is a consistent esti-
2
mator of (when ) by using Chebyshev’s
inequality. See Section 5-10 (CD Only).
1, if the ith transistor
is nondefective 7-65. Order Statistics. Let X 1 , X 2 , p , X n be a
X i µ i 1, 2 random sample of size n from X, a random variable hav-
0, if the ith transistor
ing distribution function F(x). Rank the elements in or-
is defective
der of increasing numerical magnitude, resulting in X (1) ,
X (2) , p , X (n) , where X (1) is the smallest sample element
Determine the joint probability function for X 1 and X 2 . (X (1) min{X 1 , X 2 , p , X n }) and X (n) is the largest sam-
What are the marginal probability functions for X 1 and ple element (X (n) max{X 1 , X 2 , p , X n }). X (i) is called
X 2 ? Are X 1 and X 2 independent random variables? the ith order statistic. Often the distribution of some of
7-62. When the sample standard deviation is based on the order statistics is of interest, particularly the mini-
a random sample of size n from a normal population, it mum and maximum sample values. X (1) and X (n) , respec-
can be shown that S is a biased estimator for . Spe- tively. Prove that the cumulative distribution functions
cifically, of these two order statistics, denoted respectively by
1t2 and F X 1n2 1t2 are
F X 112
E1S 2 12 1n 12
1n 22
31n 12 24
n
1t2 1 31 F1t24
F X 112
(a) Use this result to obtain an unbiased estimator for F X 1n2 1t2 3F1t24 n
of the form c n S, when the constant c n depends on the
sample size n. Prove that if X is continuous with probability density
(b) Find the value of c n for n 10 and n 25. function f (x), the probability distributions of X (1) and
Generally, how well does S perform as an estimator X (n) are
of for large n with respect to bias?

7-63. A collection of n randomly selected parts is 1t2 n31 F1t24 n 1 f 1t2
f X 11 2
measured twice by an operator using a gauge. Let X i and
Y i denote the measured values for the ith part. Assume f X 1n2 1t2 n3F1t24 n 1 f 1t2
that these two random variables are independent and
normally distributed and that both have true mean i and 7-66. Continuation of Exercise 7-65. Let X 1 , X 2 , p ,
2
variance . X n be a random sample of a Bernoulli random variable
(a) Show that the maximum likelihood estimator of 2 with parameter p. Show that
n
2
is ˆ 11 4n2 g i 1 1X i Y i 2 2 .
2
(b) Show that ˆ 2 is a biased estimator for . What P1X 1n2 12 1 11 p2 n
happens to the bias as n becomes large? P1X 112 02 1 p n
2
(c) Find an unbiased estimator for .
7-64. Consistent Estimator. Another way to measure Use the results of Exercise 7-65.
the closeness of an estimator
ˆ to the parameter is in 7-67. Continuation of Exercise 7-65. Let X 1 , X 2 , p ,
ˆ
terms of consistency. If
n is an estimator of based on X n be a random sample of a normal random variable
ˆ 2
a random sample of n observations,
n is consistent for with mean and variance . Using the results of
if Exercise 7-65, derive the probability density functions
of X (1) and X (n) .
ˆ
lim P 10
n 0 2 1
nS