95. Find the mathematical expectation of the number of points that fall when throwing a dice.
Solution. The random variable $X$ of the number of points takes the values $1,2,3,4,5,6$. We will construct its distribution law:

| Values $x_{i}$ | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| Probabilities $p_{l}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ |

Then the mathematical expectation is

$$
M(X)=1 \cdot \frac{1}{6}+2 \cdot \frac{1}{6}+3 \cdot \frac{1}{6}+4 \cdot \frac{1}{6}+5 \cdot \frac{1}{6}+6 \cdot \frac{1}{6}=3.5
$$

Different random variables can have the same mathematical expectation. Therefore, it is necessary to introduce another numerical characteristic to measure the degree of dispersion, the spread of values taken by the random variable $X$ around its mathematical expectation.

Consider the difference $x-m$, where $m$ is the mathematical expectation of the variable $X$.

The random variable $x-m$ is called the deviation of the variable from its mathematical expectation.

The variance of a random variable $X$ is the number

$$
D(X)=M\left[(x-m)^{2}\right]
$$

In other words, the variance is the mathematical expectation of the square of the deviation of the random variable from its mathematical expectation.