Example 1. A discrete random variable $X$ is given by the distribution law

| $X$ | 1 | 2 | 3 | 4 | 5 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |

Write the distribution law of the random variable $Y=X^{3}$.
Solution. The random variable $Y$ takes values equal to the cubes of the values of the variable $X$, with the same probabilities as the random variable $X$. The cubes of the values of the random variable $X$ are:

$$
\begin{gathered}
y_{1}=x_{1}^{3}=1, y_{2}=x_{2}^{3}=8, y_{3}=x_{3}^{3}=3^{3}=27 \\
y_{4}=x_{4}^{3}=4^{3}=64, y_{5}=x_{5}^{3}=5^{3}=125
\end{gathered}
$$

The distribution law of the random variable $Y=X^{3}$ is given by the following table:

| $Y=X^{3}$ | 1 | 8 | 27 | 64 | 125 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |