Page 328 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 311

Suppose that a sample X 1 , ..., X n is taken from population X, each X j consisting of
testing X j components when the first failure occurs.
(a) Determine the MLE of p.
(b) Determine the MLE of P(X > 9), the probability that the component will not
fail in nine trials. Note:
9
X
P X > 9†ˆ 1 p†p k 1 :
kˆ1
9.21 The pdf of a population X is given by

2x
8
< ; for 0 x ;
f X x; †ˆ 2
0; elsewhere:
:
Based on a sample of size n:
(a) Determine the MLE and ME for .
(b) Which one of the two is the better estimator?
9.22 Assume that X has a shifted exponential distribution, with
f X x; a†ˆ e a x ; x a:

On the basis of a sample of size n from X, determine the MLE and ME for a.
9.23 Let X 1 , X 2 ,..., X n be a sample of size n from a uniform distribution
1 1

f x; †ˆ 1; for 2 x ‡ ;
2
0; elsewhere:
Show that every statistic h(X 1 , . .., X n ) satisfying
1 1
X n† h X 1 ; ... ; X n † X 1† ‡
2 2
is an MLE for , where X (j) is the jth-order statistic. Determine an MLE for when

the observed sample values are (1.5, 1.4, 2.1, 2.0, 1.9, 2.0, 2.3), with n ˆ 7.
9.24 Using the 214 measurements given in Example 9.11 (see Table 9.1), determine the
MLE for in the exponential distribution given by Equation (9.70).
9.25 Let us assume that random variable X in Problem 8.2(j) is Poisson distributed.
Using the 58 sample values given (see Figure 8.6), determine the MLE and ME for
the mean number of blemishes.
9.26 The time-to-failure T of a certain device has a shifted exponential distribution;
that is,

t t 0 †
e ; for t t 0 ;
f T t; t 0 ; †ˆ
0; elsewhere:

TLFeBOOK