Example 2. The distribution law of a discrete random variable is given by the table

| $X$ | -4 | -2 | 0 | 2 | 4 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.1 | 0.2 | 0.15 | 0.25 | 0.3 |

Write the distribution laws of the random variables $3X$, $X / 2$. Find the mathematical expectations of the random variables $X$, $3X$, $X / 2$.
Solution. Let's write down the distribution laws of the random variables $3X$ and $X/2$ using tables:

| $3X$ | -12 | -6 | 0 | 6 | 12 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.1 | 0.2 | 0.15 | 0.25 | 0.3 |

| $X/2$ | -2 | -1 | 0 | 1 | 2 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.1 | 0.2 | 0.15 | 0.25 | 0.3 |

Using formula (2.4.3), we calculate the mathematical expectations of these variables:

\[
\begin{aligned}
& M(X) = -4 \cdot 0.1 + (-2) \cdot 0.2 + 0 \cdot 0.15 + 2 \cdot 0.25 + 4 \cdot 0.3 = 0.9 \\
& M(3X) = -12 \cdot 0.1 + (-6) \cdot 0.2 + 0 \cdot 0.15 + 6 \cdot 0.25 + 12 \cdot 0.3 = 2.7 \\
& M(X/2) = -2 \cdot 0.1 + (-1) \cdot 0.2 + 0 \cdot 0.15 + 1 \cdot 0.25 + 2 \cdot 0.3 = 0.45
\end{aligned}
\]

Remark. The mathematical expectations of the random variables $3X$ and $X/2$ can also be calculated using equality (2.4.11) if the mathematical expectation of the variable $X$ is known:

\[
\begin{aligned}
& M(3X) = 3M(X) = 3 \cdot 0.9 = 2.7 \\
& M\left(\frac{1}{2}X\right) = \frac{1}{2}M(X) = \frac{1}{2} \cdot 0.9 = 0.45
\end{aligned}
\]