Page 325 - Fundamentals of Probability and Statistics for Engineers

308 Fundamentals of Probability and Statistics for Engineers

(a) Determine the pdfs of X (1) and X (10) .
(b) Find the probabilities P[X 1) > 0:5] and P[X 10) 0:5].
(c) Determine EfX 1) g and EfX 10) g.
9.6 A sample of size n is taken from a population X with pdf

x
e ; for x 0;
f X x†ˆ
0; elsewhere:
Determine the probability density function of statistic X . (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,
1 x=
f X x; †ˆ e ; x 0:

We propose two estimators for in the forms

^
1 ˆ X ˆ X 1 ‡ X 2 ;
2
^
2 ˆ 4 X 1 X 2 † 1=2 :

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

^
M 1 ˆ X ˆ X 1 ‡ X 2 ;
2
^
M 2 ˆ X 1 ‡ 2X 2 :
3
Which is the better estimator?
(b) Consider an estimator for m in the form
^
M ˆ aX 1 ‡ 1 a†X 2 ; 0 a 1:
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 ,.. . , X n ) to be 1 if good and 0 if defective. Then, a good estimator for
p, ^ , P is

1
^
P ˆ 1 X ˆ 1 X 1 ‡ ‡ X n †:
n

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