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232 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

Suppose that this estimator was applied to the following situation: n items are selected
at random from a production line, and each item is judged as either defective (in which case
n
we set x 1) or nondefective (in which case we set x 0). Then g i 1 x i is the number of
i
i
p ˆ
defective units in the sample, and is the sample proportion defective. The parameter p is
the population proportion defective; and it seems intuitively quite reasonable to use as
p ˆ
an estimate of p.
Although the interpretation of the likelihood function given above is confined to the dis-
crete random variable case, the method of maximum likelihood can easily be extended to a
continuous distribution. We now give two examples of maximum likelihood estimation for
continuous distributions.

EXAMPLE 7-7 Let X be normally distributed with unknown and known variance 2 . The likelihood
function of a random sample of size n, say X 1 , X 2 , p , X n , is

n 1 1 n
2
2
1
L1 2 q e 1x i 2 12 2 2 2 n 2 e 11 2 2 a x i 2 2
i 1
i 1 12 12 2
Now
n
2 1
2
ln L1 2 1n 22 ln12 2 12 2 a 1x 2 2
i
i 1
and
d ln L1 2 n
2 1
1 2 a 1x 2
d i 1 i
Equating this last result to zero and solving for yields
n
a X i
i 1
ˆ X
n
Thus the sample mean is the maximum likelihood estimator of . Notice that this is identical
to the moment estimator.

EXAMPLE 7-8 Let X be exponentially distributed with parameter . The likelihood function of a random
sample of size n, say X , X , p , X , is
1
2
n
n n
L1 2 q e x i e i 1
n a x i
i 1
The log likelihood is
n
ln L1 2 n ln a x i
i 1