(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,...,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.
