Page 298 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 281
^
^
Estimates 1 and 2 of 1 ˆ m and 2 ˆ 2 based on the sample values given
by Table 8.1 are, following Equations (9.64) and (9.65),
200
1 X
^
1 ˆ x j  70;
200
jˆ1
200
^
^
2
2 ˆ 1 X x j 1 †  4;
200
jˆ1
where x j , j ˆ 1, 2, . . . , 200, are sample values given in Table 8.1.
Example 9.10. Problem: consider the binomial distribution
k
p X k; p†ˆ p 1 p† 1 k ; k ˆ 0; 1: 9:66†
Estimate parameter p based on a sample of size n.
Answer: the method of moments suggests that we determine the estimator for
^
p, P, by equating 1 to M 1 ˆ X . Since
1 ˆ EfXgˆ p;

we have

^
P ˆ X: 9:67†
The mean of P ^ is
n
1 X
^
EfPgˆ EfX j gˆ p: 9:68†
n
jˆ1
Hence it is an unbiased estimator. Its variance is given by

2 p 1 p†
^
varfPgˆ varfXgˆ ˆ : 9:69†
n n
It is easy to derive the CRLB for this case and show that P ^ 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

f t; †ˆ e t ; t 0; 9:70†

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

TLFeBOOK