1. Choose 5 numbers (repetition allowed) from $\{1,2, \cdots, 100\}$. Then the expected number of composite numbers taken is
$$
-1 . \frac{37}{10} \text {. }
$$
In the set $\{1,2, \cdots, 100\}$, there are 74 composite numbers. Let $\xi$ be the number of composite numbers drawn. Then
$$
P(\xi=i)=\mathrm{C}_{5}^{i}\left(\frac{74}{100}\right)^{i}\left(\frac{26}{100}\right)^{5-i}(0 \leqslant i \leqslant 5) .
$$

Therefore, $\xi$ follows a binomial distribution.
Thus, $\mathrm{E} \xi=5 \times \frac{74}{100}=\frac{37}{10}$.