解答
1
$$\mathbb{E}\left(1\right) = \sum_{k=1}^m 1\cdot q_k = 1$$
2
$$\mathbb{E}\left(X\right) = \sum_{k=1}^m k\mathbb{P}\left(X=k\right) = \sum_{k=1}^mkq_k$$
3
$$\mathbb{E}\left(Y_k\right) = \sum_{k^{\prime}=1}^m \mathbb{I}\left(X=k\right) \mathbb{P}\left(X=k^{\prime}\right)=\mathbb{P}\left(X=k\right) = q_k$$
4
EX1. Covariance より、
$$
\begin{aligned}
\mathrm{Cov}\left(Y_k,Y_l\right) &= \mathbb{E}\left(Y_kY_l\right) - \mathbb{E}\left(Y_k\right)\mathbb{E}\left(Y_l\right)\\
&= \sum_{k^{\prime},l^{\prime}}\mathbb{I}\left(X=k,X=l\right)\mathbb{P}\left(X=k^{\prime},X=l^{\prime}\right) - q_kq_l\\
&=\delta_{kl}q_k - q_kq_l
\end{aligned}
$$