192. Given the list of possible values of a discrete random variable $X: x_{1}=-1, x_{2}=0, x_{3}=1$, and the known mathematical expectations of this variable and its square: $M(X)=0.1, M\left(X^{2}\right)=0.9$. Find the probabilities
$p_{1}, p_{2}, p_{3}$, corresponding to the possible values $x_{1}$, $x_{2}, x_{3}$.
Solution. Using the fact that the sum of the probabilities of all possible values of $X$ is one, and also taking into account that $M(X)=0.1, M\left(X^{2}\right)=0.9$, we can form the following system of three linear equations with respect to the unknown probabilities:

$$
\begin{gathered}
p_{1}+p_{2}+p_{3}=1,(-1) p_{1}+0 \cdot p_{2}+1 \cdot p_{3}=0.1, \\
(-1)^{2} p_{1}+0^{2} \cdot p_{2}+1^{2} \cdot p_{3}=0.9 .
\end{gathered}
$$

Solving this system, we find the required probabilities: $p_{1}=0.4$, $p_{2}=0.1, p_{3}=0.5$.