Fundamentals of Probability and Statistics for Engineers

9.1.1 Sample M ean

The statistic

is called the sample mean of population X. Let the population mean and
variance be, respectively,

The mean and variance ofX, the sample mean, are easily found to be

and, owing to independence,

which is inversely proportional to sample size n. As n increases, the variance ofX
decreases and the distribution ofX becomessharply peaked at. Hence,
it is intuit ively clear tha t sta tisticX provides a good procedure for estimating
population mean m. This is another statement of the law of large numbers that
was discussed in Example 4.12 (page 96) and Example 4.13 (page 97).
SinceX is a sum of independent random variables, its distribution can also be
determined either by the use of techniques developed in Chapter 5 or by means of
the method of characteristic functions given in Section 4.5. We further observe
that, on the basis of the central limit theorem (Section 7.2.1), sample meanX
approaches a normal distribution as. M ore precisely, random variable

approaches N(0, 1) as

Parameter Estimation 261

Xˆ

1

n

Xn

iˆ 1

Xi … 9 : 3 †

EfXgˆm;

varfXgˆ^2 :

)

… 9 : 4 †

EfXgˆ

1

n

Xn

iˆ 1

EfXigˆ

1

n

…nm†ˆm; … 9 : 5 †

varfXgˆEf…Xm†^2 gˆE

1

n

Xn

iˆ 1

…Xim†

(^8) "# 2
<
:

9

=

;

ˆ

1

n^2

…n^2 †ˆ

^2

n

;

… 9 : 6 †

EfXgˆm

n!1

…Xm†



n^1 =^2

 1

n!1.