(a) Let $X$ and $Y$ be two independent discrete random variables. Derive a formula for expressing the distribution of the sum $S = X + Y$ in terms of the distributions of $X$ and of $Y$.
(b) Use your formula in part (a) to compute the distribution of $S = X +Y$ if $X$ and $Y$ are both discrete and uniformly distributed on $\{1,\dots,K\}.$
(c) Suppose now $X$ and $Y$ are continuous random variables with densities $f$ and $g$ respectively ($X,Y$ still independent). Based on part (a) and your understanding of continuous random variables, give an educated guess for the formula of the density of $S = X +Y$ in terms of $f$ and $g$.
(d) Use your formula in part (c) to compute the density of $S$ if $X$ and $Y$ have both uniform densities on $[0, a]$.
(e) Show that if $X$ and $Y$ are independent normally distributed variables, then $X +Y$ is also a normally distributed variable.
Added from comment: What I think I have figured out so far is:
a) distribution of $S$ is $\text{mean}(S) = \text{mean}(X) + \text{mean}(Y)$ and $\text{stddev}(S) = \sqrt{\text{stddev}(X)^2 + \text{stddev}(Y)^2}$
b) stuck a little on b... help please