# Question

Suppose the possible values of X are {xi}, the possible values of Y are {yj}, and the possible values of X + Y are {zk}. Let Ak denote the set of all pairs of indices (i, j) such that xi + yj = zk; that is,

Ak = {(i, j): xi + yj = zk}.

(a) Argue that

(b) Show that

(c) Using the formula from part (b), argue that

(d) Show that

(e) Prove that

E[X + Y] = E[X] + E[Y]

Ak = {(i, j): xi + yj = zk}.

(a) Argue that

(b) Show that

(c) Using the formula from part (b), argue that

(d) Show that

(e) Prove that

E[X + Y] = E[X] + E[Y]

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