Introduction to Probability and Statistics for Engineers and Scientists

234 Chapter 7: Parameter Estimation

pˆi,i=1,...,m, as preliminary estimates of thepi. Now, letnf be the number of errors
that are found by at least one proofreader. Becausenf/Nis the fraction of errors that are
found by at least one proofreader, this should approximately equal 1−

∏m
i= 1 (1−pi), the
probability that an error is found by at least one proofreader. Therefore, we have

nf

N

≈ 1 −

∏m

i= 1

(1−pi)

suggesting thatN≈Nˆ, where

Nˆ = nf

1 −

∏m

i= 1 (1−pˆi)

(7.2.1)

With this estimate ofN, we can then reset our estimates of thepiby using

pˆi=

ni

Nˆ

, i=1,...,m (7.2.2)

We can then reestimateNby using the new value (7.2.1). (The estimation need not stop
here; each time we obtain a new estimateNˆofNwe can use (7.2.2) to obtain new estimates
of thepi, which can then be used to obtain a new estimate ofN, and so on.) ■

EXAMPLE 7.2c Maximum Likelihood Estimator of a Poisson ParameterSuppose X 1 ,...,Xn
are independent Poisson random variables each having meanλ. Determine the maxi-
mum likelihood estimator ofλ.

SOLUTION The likelihood function is given by

f(x 1 ,...,xn|λ)=

e−λλx^1

x 1!

···

e−λλxn

xn!

=

e−nλλn^1 xi

x 1 !...xn!

Thus,

logf(x 1 ,...,xn|λ)=−nλ+

∑n

1

xilogλ−logc

wherec=

∏n

i= 1 xi!does not depend onλ, and

d

dλ

logf(x 1 ,...,xn|λ)=−n+

∑n

1

xi

λ