1.2 The mean, variance, and standard deviation

The mean, variance, and standard deviation

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. $f\left(x\right)$ of the discrete type and space R = $\left(b_1,b_2,b_3\mathrm{,...}\right)$ , then $E\left(X\right)={\displaystyle \sum _{R}xf\left(x\right)=b_{1}f\left(b_1\right)+b_{2}f\left(b_2\right)+b_3}f\left(b_3\right)+\mathrm{...}$ is the weighted average of the numbers belonging to R , where the weights are given by the p.d.f. $f\left(x\right)$ .

We call $E\left(X\right)$ the mean of X (or the mean of the distribution ) and denote it by $\mu$ . That is, $\mu =E\left(X\right)$ .

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.

Variance

It was denoted that the mean $\mu =E\left(X\right)$ is the centroid of a system of weights of measure of the central location of the probability distribution of X . A measure of the dispersion or spread of a distribution is defined as follows:

If $u\left(x\right)={\left(x-\mu \right)}^2$ and $E\left[{\left(X-\mu \right)}^2\right]$ exists, the variance , frequently denoted by ${\sigma }^2$ or $Var\left(X\right)$ , of a random variable 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

Moments of the distribution

Let r be a positive integer. If $$E\left(X^r\right)={\displaystyle \sum _R{x}^{r}f\left(x\right)}$$ exists, it is called the r th moment of the distribution about the origin. The expression moment has its origin in the study of mechanics.

In addition, the expectation $$E\left[{\left(X-b\right)}^r\right]={\displaystyle \sum _R{x}^{r}f\left(x\right)}$$ is called the r th moment of the distribution about b . For a given positive integer r.

$$E\left[{\left(X\right)}_r\right]=E\left[X\left(X-1\right)\left(X-2\right)\cdot \cdot \cdot \left(X-r+1\right)\right]$$ is called the r th factorial moment .

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 $E\left(X\right)$ and $E\left[X\left(X-1\right)\right]$ . Then $${\sigma }^2=E\left[X\left(X-1\right)\right]+E\left(X\right)-{\left[E\left(X\right)\right]}^2,$$ since using the distributive property of E , this becomes $${\sigma }^2=E\left(X^2\right)-E\left(X\right)+E\left(X\right)-{\left[E\left(X\right)\right]}^2=E\left(X^2\right)-{\mu }^2.$$

The symbol $\overline{x}$ represents the mean of the empirical distribution . It is seen that $\overline{x}$ is usually close in value to $\mu =E\left(X\right)$ ; thus, when $\mu$ is unknown, $\overline{x}$ will be used to estimate $\mu$ .

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

$$v={{\displaystyle \sum _{i=1}^n\left(x_i-\overline{x}\right)}}^2\frac{1}{n}={\displaystyle \sum _{i=1}^n{x}_i^2\frac{1}{n}-{\overline{x}}^2}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i^2-{\overline{x}}^2}.$$

This last statement is true because, in general, $${\sigma }^2=E\left(X^2\right)-{\mu }^2.$$