Answer

We use the formulae from iterated expectation to calculate these.

\[ \begin{alignat}{2} P_{Y}(y) &= \begin{cases} \frac{1}{3} &y = 1\newline \frac{2}{3} &y = 2 \end{cases}\newline E[X] &= E[E[X \vert Y]] = \sum_{y}E[X \vert Y]P(Y)\newline &= 90 \times \frac{1}{3} + 60 \times \frac{2}{3}\newline Var(X) &= E[Var(X \vert Y)] + Var(E[X \vert Y])\newline &= \sum_{y}Var(X \vert Y)P(Y) + ((90-E[E[X \vert Y])^{2}\frac{1}{3} + (60-E[E[X \vert Y]])^{2}\frac{2}{3})\newline &= \frac{650}{3} \end{alignat} \]