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(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

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    What I think I have figured out so far is: a) distribution of S is Smean = mean of X + mean of Y and Sstanddev = sqrt(stddevX^2 + stddevY^2) b) stuck a little on b... help please2011-11-17
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    For a), use $P[X+Y=k]=\sum_i P[X=k-i , Y=i] = \sum_i P[X=k-i]P[Y=i]$. For b), you know the pmf's of $X$ and $Y$, so just substitute into the formula from a).2011-11-17
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    hmm...i'm confused as to where you got that equation/expression from and also are for part b) is pmf is 1/k because of uniform distrib2011-11-17

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