Applied Statistics and Probability for Engineers

7-5 SAMPLING DISTRIBUTIONS OF MEANS 245

MIND-EXPANDING EXERCISES

7-61. A lot consists of Ntransistors, and of these M

(MN) are defective. We randomly select two transis-

tors without replacement from this lot and determine

whether they are defective or nondefective. The ran-

dom variable

Determine the joint probability function for X 1 and X 2.

What are the marginal probability functions for X 1 and

X 2? Are X 1 and X 2 independent random variables?

7-62. When the sample standard deviation is based on

a random sample of size nfrom a normal population, it

can be shown that Sis a biased estimator for . Spe-

cifically,

(a) Use this result to obtain an unbiased estimator for 

of the form cnS, when the constant cndepends on the

sample size n.

(b) Find the value of cn for n10 and n25.

Generally, how well does Sperform as an estimator

of for large nwith respect to bias?

7-63. A collection of nrandomly selected parts is

measured twice by an operator using a gauge. Let Xiand

Yidenote the measured values for the ith part. Assume

that these two random variables are independent and

normally distributed and that both have true mean iand

variance ^2.

(a) Show that the maximum likelihood estimator of ^2

is.

(b) Show that is a biased estimator for ^2. What

happens to the bias as nbecomes large?

(c) Find an unbiased estimator for ^2.

7-64. Consistent Estimator.Another way to measure

the closeness of an estimator to the parameter is in

terms of consistency. If is an estimator of based on

a random sample of nobservations, is consistent for

if

Thus, consistency is a large-sample property, describing

the limiting behavior of as ntends to infinity. It is

usually difficult to prove consistency using the above

definition, although it can be done from other ap-

proaches. To illustrate, show that is a consistent esti-

mator of (when )by using Chebyshev’s

inequality. See Section 5-10 (CD Only).

7-65. Order Statistics.Let X 1 , X 2 ,, Xn be a

random sample of size nfrom X, a random variable hav-

ing distribution function F(x). Rank the elements in or-

der of increasing numerical magnitude, resulting in X(1),

X(2),, X(n), where X(1)is the smallest sample element

(X(1)min{X 1 , X 2 ,, Xn}) and X(n)is the largest sam-

ple element (X(n)max{X 1 , X 2 ,, Xn}). X(i)is called

the ith order statistic. Often the distribution of some of

the order statistics is of interest, particularly the mini-

mum and maximum sample values. X(1)and X(n), respec-

tively. Prove that the cumulative distribution functions

of these two order statistics, denoted respectively by

and are

Prove that if Xis continuous with probability density

function f(x), the probability distributions of X(1)and

X(n)are

7-66. Continuation of Exercise 7-65. Let X 1 , X 2 ,,

Xnbe a random sample of a Bernoulli random variable

with parameter p. Show that

Use the results of Exercise 7-65.

7-67. Continuation of Exercise 7-65. Let X 1 , X 2 ,,

Xnbe a random sample of a normal random variable

with mean and variance ^2. Using the results of

Exercise 7-65, derive the probability density functions

of X(1)and X(n).

p

P 1 X 112  02  1 pn

P 1 X 1 n 2  12  1  11 p 2 n

p

fX 1 n 21 t 2 n 3 F 1 t 24 n^1 f 1 t 2

fX 11 21 t 2 n 31 F 1 t 24 n^1 f 1 t 2

FX 1 n 21 t 2  3 F 1 t 24 n

FX 1121 t 2  1  31 F 1 t 24 n

FX 1121 t 2 FX 1 n 21 t 2

p

p

p

p

^2    

X

ˆn

limnS P 10

 ˆn^0  2  1

ˆn

ˆn

ˆ

ˆ^2

ˆ^2  114 n 2 gni 1 1 XiYi 22



E 1 S 2  121 n 12
1 n 22
31 n 1224

Xiμ

1, if the ith transistor

is nondefective

0, if the ith transistor

is defective

i1, 2

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