Toss a fair die and let $N$ be the outcome. Next, throw a coin $N$ times. Then, we denote $S$ the number of heads gotten. What is the expectation and the variance of $S$?
I tried defining the variable $S$ as $S = N*S_i$, where $S_i$ is toss $i$.
Then by independence of $N$ and $S_i I$ got the expectation as follows: $E[S] = E[N]E[S_i] = 3.5 * 0.5 = 1.75$.
For the variance also by independence: $$Var[S] = (E[N]^2) * Var[S_i] + (E[S_i]^2) Var[N] + Var[S_i]Var[N] \\= (3.5^2) * 0.5(1-0.5) + (0.5^2) * (35/12) + (35/12) * 0.5(1-0.5) = 4 25/48.$$ However, the solution model says that it should be $77/48$, so I think i'm tackling the problem the wrong way.