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7-3 METHODS OF POINT ESTIMATION 231

Definition

Suppose that X is a random variable with probability distribution f(x; ), where is
a single unknown parameter. Let x , x , p , x be the observed values in a random
2
1
n
sample of size n. Then the likelihood function of the sample is
L1 2 f 1x ; 2 f 1x ; 2 p f 1x ; 2 (7-5)
1
2
n
Note that the likelihood function is now a function of only the unknown parameter .
The maximum likelihood estimator of is the value of that maximizes the like-

lihood function L().

In the case of a discrete random variable, the interpretation of the likelihood function is

clear. The likelihood function of the sample L( ) is just the probability
P1X x , X x , p , X x 2
n
2
1
2
1
n
That is, L( ) is just the probability of obtaining the sample values x , x , p 1 2 , x . Therefore, in
n
the discrete case, the maximum likelihood estimator is an estimator that maximizes the prob-
ability of occurrence of the sample values.
EXAMPLE 7-6 Let X be a Bernoulli random variable. The probability mass function is
x 1 x
p 11 p2 , x 0, 1
f 1x; p2 e
0, otherwise
where p is the parameter to be estimated. The likelihood function of a random sample of size
n is

L1 p2 p 11 p2 1 x 1 x 2 1 x 2 p x n 1 x n
x 1
p 11 p2
p 11 p2
n n n
q p 11 p2 1 x i p i 1 11 p2 n a x i
a x i
x i
i 1
i 1
p ˆ
p ˆ
We observe that if maximizes L(p), also maximizes ln L(p). Therefore,
n n
ln L1 p2 a a x b ln p an a x b ln 11 p2
i
i
i 1 i 1
Now
n n
a x i an a x b
i
d ln L1 p2 i 1 i 1

dp p 1 p
n
Equating this to zero and solving for p yields p ˆ 11 n2 g i 1 x i . Therefore, the maximum
likelihood estimator of p is
1 n
ˆ
n a
P X i
i 1