Suppose the possible values of X are xi the possible value o

Suppose the possible values of X are {x_i}, the possible value of Y are {y_i}, and the possible values of X + Y are {z_k}. Let A_k denote the set of all possible pairs of indicies (i, j) such that x_i + y_j = z_k; that is, A_k = {(i, j) : x_i + y_j = z_k}. Argue that P{X + Y = z_k}= Sigma (i, j) A_k P{X = x_i, Y = y_j}. Show that E[X + Y] = Sigma k Sigma(i, j) A_k (x_i + y_j)P{X = x_i,Y = y_j}. Using (b), argue that E[X + Y] = Sigma i Sigma j (x_i + y_j)P{X = x_i,Y = y_j}. Show that P{X = x_i} = SigmaP{X = x_i, Y = y_j}, P{Y = y_j} = Sigma I P{X = x_i, Y = y_j}. Using these results, prove that E[X + Y] = E[X] + E[Y].

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