This is a really nice question, and while I can think of a solution to both parts, I wonder if there's a more elegant one to the latter:
A fair 6-faced dice is tossed once. In a box there are 6 coins, numbered 1 to 6. Each has $\frac{1}{i}$ chance to land Head up, where $i$ is the number of the coin. The coin is chosen based upon the result of the dice, and is thrown until it lands Head up.
We define X as the RV representing the result of the dice-throw, and Y as the RV representing the number of times the coin is tossed until (including) the toss where it lands Head up.
Find E[Y], Var(Y).
I can define Y as the summation of each $Y_i$ representing the tossing of the i-th coin (each $Y_i$ being a geometric RV), each with a probability of $\frac{1}{6}$ of happening. That means:
$$Y=\sum_{i=1}^{6}\frac{1}{6}Y_i$$
That way it is extremely easy to find E[Y]. However, Var(Y) remains difficult to find, due to the need to find so many Covariance between each $Y_i$, and thus the need to find $E[Y_i * Y_j]$ for each. Is there a simpler, more elegant way to do Var(Y) instead of computing each? Perhaps I can state something regarding them, due to their exclusiveness (I know that if $Y_i$ happened, $Y_j$ certainly didn't)? I am a bit lost regarding it, and I am rather sure there is a simpler/faster/more elegant way to do this, yet unsure of what it is. Perhaps conditional probability can come into use?
Any directions/hints/suggestions will be extremely appreciated!