12. Decimal fraction (from 8th grade. 2 points) There is a sequence of a thousand random digits. Before this sequence, a zero and a decimal point were placed. As a result, a random number $X$ was obtained, written as a decimal fraction: $X=0, x_{1} x_{2} x_{3} \ldots x_{1000}$. Find $\mathrm{E} X$.
Solution. $\quad \mathrm{E} X=0+0.1 \mathrm{E} x_{1}+0.01 \mathrm{E} x_{2}+\ldots+0.1^{k} \mathrm{E} x_{k}+\ldots+0.1^{1000} \mathrm{E} x_{1000} . \quad$ All digits take values from 0 to 9 with equal probabilities, so $\mathrm{Ex}_{k}=4.5$ for any $k$. Therefore,

$$
\mathrm{E} X=0.45+0.045+0.0045+\ldots+0. \underbrace{00 \ldots 0}_{\text {9999 zeros }} 45=0.4 \underbrace{999 \ldots .995}_{9999 \text { nines }} .
$$

Answer: $0.4 \underbrace{4999}_{9999 \text { nines }} 5$.