** MEAN and VARIANCE**

Certain mathematical expectations are so important that they have special names. In this section we consider two of them: the mean and the variance.

Mean Value

If *X* is a random variable with p.d.f. *R*=*R*, where the weights are given by the p.d.f.

We call *X* (or the mean of the distribution) and denote it by

##### REMARK:

##### Example 1

Let *X* have the p.d.f.

The mean of *X* is

The example below shows that if the outcomes of *X* are equally likely (i.e., each of the outcomes has the same probability), then the mean of *X* is the arithmetic average of these outcomes.

##### Example 2

Roll a fair die and let *X* denote the outcome. Thus *X* has the p.d.f.

which is the arithmetic average of the first six positive integers.

Variance

It was denoted that the mean *X*. A measure of the dispersion or spread of a distribution is defined as follows:

If *X* of the discrete type (or variance of the distribution) is defined by

The positive square root of the variance is called the standard deviation of *X* and is denoted by

##### Example 3

Let the p.d.f. of *X* by defined by

The mean of *X* is

To find the variance and standard deviation of *X* we first find

Thus the variance of *X* is

and the standard deviation of *X* is

##### Example 4

Let *X* be a random variable with mean *Y* is

Moreover, the variance of *Y* is

Moments of the distribution

Let *r* be a positive integer. If *r*th moment of the distribution about the origin. The expression moment has its origin in the study of mechanics.

In addition, the expectation *r*th moment of the distribution about *b*. For a given positive integer r.

*r*th factorial moment.

##### Note That:

There is another formula that can be used for computing the variance that uses the second factorial moment and sometimes simplifies the calculations.

First find the values of *E*, this becomes

##### Example 5

Let continue with example 4, it can be find that

Thus

##### REMARK:

*n*on each of

*n*observations

The symbol

Similarly, the variance of the empirical distribution can be computed. Let *v* denote this variance so that it is equal to

This last statement is true because, in general,

##### NOTE THAT:

*v*of the empirical distribution, namely

*n*, the difference between

*v*is very small. Usually, we use