210. Find the variance and standard deviation of the discrete random variable $X$, given by the distribution law:

$$
\begin{array}{ccccc}
X & -5 & 2 & 3 & 4 \\
p & 0.4 & 0.3 & 0.1 & 0.2
\end{array}
$$
Solution. The variance can be calculated from its definition, but we will use the formula

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

which leads to the goal faster.

Let's find the mathematical expectation of $X$:

$$
M(X)=-5 \cdot 0.4+2 \cdot 0.3+3 \cdot 0.1+4 \cdot 0.2=-0.3
$$

Write the distribution law of $X^{2}$:

| $X^{2}$ | 25 | 4 | 9 | 16 |
| :--- | :---: | :---: | :---: | :---: |
| $p$ | 0.4 | 0.3 | 0.1 | 0.2 |

Find the mathematical expectation of $X^{2}$:

$$
M\left(X^{2}\right)=25 \cdot 0.4+4 \cdot 0.3+9 \cdot 0.1+16 \cdot 0.2=15.3
$$

Find the required variance:

$$
D(X)=M\left(X^{2}\right)-[M(X)]^{2}=15.3-(-0.3)^{2}=15.21
$$

Find the required standard deviation:

$$
\sigma(X)=\sqrt{D(X)}=\sqrt{15.21}=3.9
$$