Fundamentals of Probability and Statistics for Engineers

Estimates based on the sample values given
by Table 8.1 are, following Equations (9.64) and (9.65),

where xj, j 1, 2,... , 200, are sample values given in Table 8.1.

Example 9.10.Problem: consider the binomial distribution

Estimate parameter p based on a sample of size n.
Answer: the method of moments suggests that we determine the estimator for
by equating 1 to M 1 X. Since

we have

The mean of is

Hence it is an unbiased estimator. Its variance is given by

It is easy to derive the CRLB for this case and show that defined by Equation
(9.67) is also efficient.

Example 9.11.Problem: a set of 214 observed gaps in traffic on a section of
Arroyo Seco Freeway is given in Table 9.1. If the exponential density function

is proposed for the gap, determine parameter from the data.

Parameter Estimation 281

^ 1 and^ 2 of 1 ˆmand 2 ˆ^2

^ 1 ˆ^1

200

X^200

jˆ 1

xj 70 ;

^ 2 ˆ^1

200

X^200

jˆ 1

…xj^ 1 †^2  4 ;

ˆ

pX…k;p†ˆpk… 1 p†^1 k; kˆ 0 ; 1 : … 9 : 66 †

p,P^, ˆ

1 ˆEfXgˆp;

P^ˆX: … 9 : 67 †

P^

EfP^gˆ

1

n

Xn

jˆ 1

EfXjgˆp: … 9 : 68 †

varfP^gˆvarfXgˆ

^2

n

ˆ

p… 1 p†

n

: … 9 : 69 †

P^

f…t;†ˆet; t 0 ; … 9 : 70 †