Page 327 - Fundamentals of Probability and Statistics for Engineers

310 Fundamentals of Probability and Statistics for Engineers

9.17 Using the maximum likelihood method and the moment method, determine the
^
respective estimators of and compare their asymptotic variances for the
following two cases:
(a) Case 1:
" 2 #
1 x m†
f x; †ˆ 1=2 exp 2 ; where m is a known constant:
2 † 2
(b) Case 2:
8
1 1
> 2
< exp ln x ; for x > 0;
f x; †ˆ x 2 † 1=2 2
>
:
0; elsewhere:
9.18 Consider each distribution in Problem 9.14.
(a) Determine an ME for on the basis of a sample of size n by using the first-
order moment equation. Determine its asymptotic efficiency (i.e. its efficiency
as n !1 ). (Hint: use the second of Equations (9.62) for the asymptotic
variance of ME.)
(b) Determine the MLE for .
9.19 The number of transistor failures in an electronic computer may be considered as a
random variable.
(a) Let X be the number of transistor failures per hour. What is an appropriate
distribution for X? Explain your answer.
(b) The numbers of transistor failures per hour for 96 hours are recorded in Table
9.3. Estimate the parameter(s) of the distribution for X based on these data by
using the method of maximum likelihood.

Table 9.3 Data for Problem 9.19
Hourly failures (No.) Hours (No.)
0 59
1 27
2 9
3 1
>3 0
Total ˆ 96

(c) A certain computation requires 20 hours of computing time. Use this model
and find the probability that this computation can be completed without a
computer breakdown (a breakdown occurs when two or more transistors fail).
9.20 Electronic components are tested for reliability. Let p be the probability of an
electronic component being successful and 1 p be the probability of component
failure. If X is the number of trials at which the first failure occurs, then it has the
geometric distribution

p X k; p†ˆ 1 p†p k 1 ; k ˆ 1; 2; .. . :

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