After doing part (a), you find a general form for the $\displaystyle M_{x}(t) = pe^t(1-(1-p)e^t)^{-1}$.

I did what utopiaNow did for part (b), and using the work from (a), I used p = 1/6

Part (c) is due to the fact that the geometric r.v. is memoryless.

Part (d), I multiplied $\displaystyle M_{X} * M_{Y-X} = M_{X+Y-X} = M_{Y}$ to find the mgf of $\displaystyle M_{Y}$. To find the $\displaystyle E(Y)$, I found the derivative of $\displaystyle M_{Y}$.

I'm not quite sure about my work for (d), but I think that's the gist of it... try it out