Example 5. A symmetric coin is tossed 4 times. The random variable $X$ - "the number of heads appearing in these tosses". Find the numerical characteristics of the random variable $X: M(X), D(X), \sigma(X)$.
Solution. The discrete random variable $X$ can take five values: $x_{1}=0, x_{2}=1, x_{3}=2, x_{4}=3, x_{5}=4$.

The distribution law of the random variable $X$ can be given by the table

| $X$ | 0 | 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | $1 / 16$ | $4 / 16$ | $6 / 16$ | $4 / 16$ | $1 / 16$ |

We find the mathematical expectation $M(X)$:

$$
M(X)=0 \cdot 1 / 16+1 \cdot 4 / 16+2 \cdot 6 / 16+3 \cdot 4 / 16+4 \cdot 1 / 16=2
$$

The distribution law of the random variable $(X-M(X))^{2}$ is as follows:

| $(X-M(X))^{2}$ | $(0-2)^{2}$ | $(1-2)^{2}$ | $(2-2)^{2}$ | $(3-2)^{2}$ | $(4-2)^{2}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | $1 / 16$ | $4 / 16$ | $6 / 16$ | $4 / 16$ | $1 / 16$ |

We calculate the variance $D(X)$ and the standard deviation $\sigma(X)$:

$$
\begin{gathered}
D(X)=4 \cdot 1 / 16+1 \cdot 4 / 16+0 \cdot 6 / 16+1 \cdot 4 / 16+4 \cdot 1 / 16=1 \\
\sigma(X)=\sqrt{D(X)}=1
\end{gathered}
$$