Page 312 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 295

The standardized random variable U, defined by
p 
5 X m†
U ˆ ; 9:123†
3
is then N(0, 1) and it has pdf
1 u =2
2
f U u†ˆ 1=2 e ; 1 < u < 1: 9:124†
2 †

Suppose we specify that the probability of U being in interval ( u 1 , u 1 ) is equal
to 0.95. From Table A.3 we find that u 1 ˆ 1.96 and
Z 1:96
P 1:96 < U < 1:96†ˆ f U u†du ˆ 0:95; 9:125†
1:96
or, on substituting Equation (9.123) into Equation (9.125),

P X 2:63 < m < X ‡ 2:63†ˆ 0:95; 9:126†
and, using Equation (9.122), the observed interval is

P 0:81 < m < 4:45†ˆ 0:95: 9:127†

Equation (9.127) gives the desired result but it must be interpreted carefully.
The mean m, although unknown, is nevertheless deterministic; and it either lies
in an interval or it does not. However, we see from Equation (9.126) that the
interval is a function of statistic X . Hence, the proper way to interpret Equa-
tions (9.126) and (9.127) is that the probability of the random interval
(X 2.63, X ‡ 2.63) covering the distribution’s true mean m is 0.95, and Equa-
tion (9.127) gives the observed interval based upon the given sample values.
Let us place the concept illustrated by the example above in a more general
and precise setting, through Definition 9.2.

Definition 9.2. Suppose that a sample X 1 , X 2 ,..., X n is drawn from a popula-

tion having pdf f x; ),
being the parameter to be estimated. Further suppose
that L 1 (X 1 ,. . ., X n ) and L 2 (X 1 ,..., X n ) are two statistics such that L 1 < L 2 with
probability 1. The interval (L 1 , L 2 ) is called a [100(1 )]% confidence interval
for if L 1 and L 2 can be selected such that
P L 1 < < L 2 †ˆ 1 : 9:128†
Limits L 1 and L 2 are called, respectively, the lower and upper confidence limits
for , and 1 is called the confidence coefficient. The value of 1 is
generally taken as 0.90, 0.95, 0.99, and 0.999.

TLFeBOOK