Page 100 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 83

Answer: for this problem, it is reasonable to expect that errors introduced in
the making of a three-foot tape measure again are accountable for inaccuracies
in the three-yard tape measures. It is then reasonable to assume that the
coefficient of variation v ˆ /m is constant for tape measures of all lengths
manufactured by this company. Thus
0:03
v ˆ ˆ 0:01;
3
and the standard deviation for a three-yard tape measures is 0:01 9 feet) ˆ
0:09 feet.
This example illustrates the fact that the coefficient of variation is often
used as a measure of quality for products of different sizes or different weights.
In the concrete industry, for example, the quality in terms of concrete strength
is specified by a coefficient of variation, which is a constant for all mean
strengths.
Central moments of higher order reveal additional features of a distribution.
The coefficient of skewness, defined by

3

1 ˆ 4:11†
3
gives a measure of the symmetry of a distribution. It is positive when a uni-
modal distribution has a dominant tail on the right. The opposite arrangement
produces a negative
1 . It is zero when a distribution is symmetrical about the
mean. In fact, a symmetrical distribution about the mean implies that all odd-
order central moments vanish.
The degree of flattening of a distribution near its peaks can be measured by
the coefficient of excess, defined by

4

2 ˆ 3: 4:12†
4
A positive
2 implies a sharp peak in the neighborhood of a mode in a unimodal
distribution, whereas a negative
2 implies, as a rule, a flattened peak. The
significance of the number 3 in Equation (4.12) will be discussed in Section 7.2,
when the normal distribution is introduced.

4.1.3 CONDITIONAL EXPECTATION

We conclude this section by introducing a useful relation involving conditional
expectation. Let us denote by EfXjYg that function of random variable Y for

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