most recent 30 from http://mathoverflow.net 2013-05-24T22:37:40Z http://mathoverflow.net/feeds/question/81156 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81156/probability-discrete-random-variables Herman Chong 2011-11-17T08:53:22Z 2011-11-17T08:58:47Z

<p>(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 .</p> <p>(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}.</p> <p>(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.</p> <p>(d) Use your formula in part (c) to compute the density of S if X and Y have both uniform densities on [0, a].</p> <p>(e) Show that if X and Y are independent normally distributed variables, then X +Y is also a normally distributed variable.</p>