Page 278 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 261
9.1.1 SAMPLE MEAN

The statistic

n
1 X
X ˆ X i 9:3†
n
iˆ1
is called the sample mean of population X. Let the population mean and
variance be, respectively,

)
EfXgˆ m;
9:4†
2
varfXgˆ :
The mean and variance of X , the sample mean, are easily found to be
n
1 X 1
EfXgˆ EfX i gˆ nm†ˆ m; 9:5†
n n
iˆ1
and, owing to independence,
8 9
" # 2
< 1 n =
2 X
varfXgˆ Ef X m† gˆ E X i m†
n
iˆ1 9:6†
: ;
1 2
2
ˆ n †ˆ ;
n 2 n
which is inversely proportional to sample size n. As n increases, the variance of X
decreases and the distribution of X becomes sharply peaked at EfXgˆ m . Hence,
it is intuitively clear that statistic X 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).
Since X 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 mean X
approaches a normal distribution as n !1 . More precisely, random variable

1
X m†
n 1=2
approaches N(0, 1) as n !1.

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