Fundamentals of Probability and Statistics for Engineers

The standardized random variable U, defined by

is then N (0, 1) and it has pdf

Suppose we specify that the probability of U being in interval ( u 1 ,u 1 )isequal
to 0.95. From Table A.3 we find that u 1 1.96 and

or, on substituting Equation (9.123) into Equation (9.125),

and, using Equation (9.122), the observed interval is

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 statisticX. 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) co vering the distribution’s true mean m is 0.95, and Equa-
tion (9.127) gives the observed interval basedupon 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.

D ef inition 9. 2. Suppose that a sample X 1 ,X 2 ,...,Xn is drawn from a popula-
tion having pdf being the parameter to be estimated. Further suppose
that L 1 (X 1 ,...,Xn)andL 2 (X 1 ,...,Xn) 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

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.

Parameter Estimation 295

Uˆ



5

p

…Xm†

3

; … 9 : 123 †

fU…u†ˆ

1

… 2 †^1 =^2

eu

(^2) = 2
; 1<u< 1 : … 9 : 124 †



P… 1 : 96 <U< 1 : 96 †ˆ

Z 1 : 96

 1 : 96

fU…u†duˆ 0 : 95 ; … 9 : 125 †

P…X 2 : 63 <m<X‡ 2 : 63 †ˆ 0 : 95 ; … 9 : 126 †

P… 0 : 81 <m< 4 : 45 †ˆ 0 : 95 : … 9 : 127 †

 ‡

fx;),





P…L 1 <<L 2 †ˆ 1 : … 9 : 128 †

  

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