1703. Determine whether Chebyshev's theorem is applicable to the sequence of independent random variables $X_{1}, X_{2}, \ldots$, $X_{n}, \ldots$, if $X_{n}$ has the following distribution:

| $x$ | $-5 n$ | 0 | $5 n$ |
| :---: | :---: | :---: | :---: |
| $p(x)$ | $\frac{1}{3 n^{2}}$ | $1-\frac{2}{3 n^{2}}$ | $\frac{1}{3 n^{2}}$ |
Solution. Chebyshev's theorem is applicable to the sequence of random variables $X_{n}$ if there exists a constant $C$ such that $D X_{n}<C$ for any $n$. Let's check this:

$$
\begin{gathered}
E X_{n}=-5 n \cdot \frac{1}{3 n^{2}}+0 \cdot\left(1-\frac{2}{3 n^{2}}\right)+5 n \cdot \frac{1}{3 n^{2}}=0 ; E X_{n}^{2}= \\
=(-5 n)^{2} \cdot \frac{1}{3 n^{2}}+(0)^{2}\left(1-\frac{2}{3 n^{2}}\right)+(5 n)^{2} \cdot \frac{1}{3 n^{2}}=\frac{25}{3}+\frac{25}{3}= \\
=\frac{50}{3} ; D X_{n}=\frac{50}{3}
\end{gathered}
$$

Thus, $C$ exists (we can take $\left.\frac{50}{3}\right)$ and Chebyshev's theorem is applicable to the sequence $\left\{X_{n}\right\}$.