Applied Statistics and Probability for Engineers

7-3 METHODS OF POINT ESTIMATION 237

7-19. Consider the Poisson distribution

Find the maximum likelihood estimator of , based on a

random sample of size n.

7-20. Consider the shifted exponential distribution

When 0, this density reduces to the usual exponential

distribution. When , there is only positive probability to

the right of .

(a) Find the maximum likelihood estimator of and , based

on a random sample of size n.

(b) Describe a practical situation in which one would suspect

that the shifted exponential distribution is a plausible

model.

 

 0



f 1 x 2 e^1 x^2 , x 



f 1 x 2 

ex

x!

, x0, 1, 2,...

7-21. Let Xbe a geometric random variable with parameter

p. Find the maximum likelihood estimator of p, based on a

random sample of size n.

7-22. Let Xbe a random variable with the following proba-

bility distribution:

Find the maximum likelihood estimator of , based on a random

sample of size n.

7-23. Consider the Weibull distribution

(a) Find the likelihood function based on a random sample of

size n. Find the log likelihood.

f 1 x 2 •





a

x



b

 1

e

axb



,0 x

0 , otherwise

f 1 x 2 e

1  12 x, 0 x 1

0 , otherwise

surfaceas a function of rand , and Figure 7-5(b) is a contour plot.These plots reveal that the

log likelihood is maximized at approximately and. Many statistics com-

puter programs use numerical techniques to solve for the maximum likelihood estimates when

no simple solution exists.

7-3.3 Bayesian Estimation of Parameters (CD Only)

EXERCISES FOR SECTION 7-3

rˆ1.75 ˆ0.08

–31.94

–31.96

–31.98

–32.00

–32.02

–32.04

–32.06

–32.08

–32.10

0.087

0.085

0.083

0.081

0.079

0.077

0.0751.58

1.62

1.66

1.70

1.74

1.78

1.82

1.86

Log likelihood

λ r

(a) (b)

0.075

0.077

0.079

0.081

0.083

0.085

0.087

1.58 1.62 1.66 1.70 1.74 1.78 1.82 1.86

r

λ

–32.106

–32.092

–32.078

–32.064

–32.05

–32.036

–32.022

–32.009

–31.995

–31.997

Figure 7-5 Log likelihood for the gamma distribution using the failure time data. (a) Log likelihood surface. (b) Contour plot.

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