Applied Statistics and Probability for Engineers

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

That is, L( ) is just the probability of obtaining the sample values x 1 , x 2 ,, xn. Therefore, in

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 Xbe a Bernoulli random variable. The probability mass function is

where pis the parameter to be estimated. The likelihood function of a random sample of size

nis

We observe that if maximizes L(p), also maximizes ln L(p). Therefore,

Now

Equating this to zero and solving for pyields. Therefore, the maximum

likelihood estimator of pis

Pˆ

1

n^ a

n

i 1

Xi

pˆ (^11) n 2 gni 1 xi
d ln L 1 p 2
dp

a
n
i 1
xi
p 
an a
n
i 1
xib
1 p
ln L 1 p 2 aa
n
i 1
xib ln pana
n
i 1
xib ln 11 p 2
pˆ pˆ
q
n
i 1
pxi 11 p 21 xipa
n
ix 1 i 11 p 2 na
n
i 1
xi
L 1 p 2 px^111 p 21 x^1 px^211 p 21 x^2 p pxn 11 p 21 xn
f 1 x; p 2 e
px 11 p 21 x, x0, 1
0, otherwise
 p
P 1 X 1 x 1 , X 2 x 2 ,p, Xnxn 2

7-3 METHODS OF POINT ESTIMATION 231
Suppose that Xis a random variable with probability distribution f(x; ), where is
a single unknown parameter. Let x 1 , x 2 ,, xnbe the observed values in a random
sample of size n. Then the likelihood functionof the sample is
(7-5)
Note that the likelihood function is now a function of only the unknown parameter
The maximum likelihood estimatorof is the value of that maximizes the like-
lihood function L().

.
L 1  2 f 1 x 1 ;  2 f 1 x 2 ;  2 pf 1 xn;  2
p

Definition
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