Fundamentals of Probability and Statistics for Engineers

(a) Determine the pdfs of X(1) and X(10).

(b) Find the probabilities

(c) D etermine

9.6 A sample of size n is taken from a population X with pdf

Determine the probability density function of statisticX. (Hint: use the method of

characteristic functions discussed in Chapter 4.)

9.7 Two samples X 1 and X 2 are taken from an exponential random variable X with
unknown parameter ; that is,

We propose two estimators for in the forms

In terms of unbiasedness and minimum variance, which one is the better of the two?

9.8 Let X 1 and X 2 be a sample of size 2 from a population X with mean m and variance

(^2).
(a) Two estimators for m are proposed to be
Which is the better estimator?
(b) Consider an estimator for m in the form
Determine value a that gives the best estimator in this form.
9.9 It is known that a certain proportion, say p, of manufactured parts is defective.
From a supply of parts, n are chosen at random and are tested. Define the readings
(sample X 1 ,X 2 ,...,Xn) to be 1 if good and 0 if defective. Then, a good estimator for
is,
308 Fundamentals of Probability and Statistics for Engineers
P[X1)> 0 :5] andP[X10) 0 :5].
EfX1)gandEfX10)g.
fX…x†ˆ e
x; forx 0 ;
0 ; elsewhere:


fX…x;†ˆ
1

ex=; x 0 :

^ 1 ˆXˆX^1 ‡X^2
2
;
^ 2 ˆ^4

…X 1 X 2 †^1 =^2 :

M^ 1 ˆXˆX^1 ‡X^2
2
;
M^ 2 ˆX^1 ‡^2 X^2
3
:
M^ˆaX 1 ‡… 1 a†X 2 ; 0 a 1 :
p,P^
P^ˆ 1 Xˆ 1 ^1
n
…X 1 ‡‡Xn†: