Fundamentals of Probability and Statistics for Engineers

Suppose that a sample X 1 ,...,Xn is taken from population X, each Xj consisting of

testing Xj 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. N ote:

9.21 The pdf of a population X is given by

Based on a sample of size n:

(a) D etermine the M LE and M E for.

(b) Which one of the two is the better estimator?

9.22 Assume that X has a shifted exponential distribution, with

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 ,...,Xn be a sample of size n from a uniform distribution

Show that every statistic h(X 1 ,...,Xn) satisfying

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 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.
U sing the 58 sample values given (see F igure 8.6), determine the M LE and M E for
the mean number of blemishes.

9.26 The time-to-failure T of a certain device has a shifted exponential distribution;
that is,

Parameter Estimation 311

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