1. (10 pts) Let X, Y Z 0 be two independent nonnegative integer-valued random variables with...

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1. (10 pts) Let X, Y Z 0 be two independent nonnegative integer-valued random variables with probability mass functions px (x), py(y) respectively. Consider Z = X + Y, and its probability mass function pz(z). (a) (5 pts) Prove that Pz(z) = Σpx(x)py (z − x). (Hint: Show that the event "Z Yzx" for x = 0,..., z.) = Z = x=0 z" is the union of the disjoint events "X = x AND (b) (5 pts) Use part (a) to prove that if X~ Binom(n, a), Y~ Binom(m, a), then Z~ Binom(n + m, a). x=0 (Hint: First prove that Σ²_o (") (m₂) = (n+m). This can be proved by induction, or by considering how many ways one can choose z items among n + m items, where the items are divided into two groups of sizes n and m. Here we assume = 0 if x <0 or x > n.) 1. (10 pts) Let X, Y Z 0 be two independent nonnegative integer-valued random variables with probability mass functions px (x), py(y) respectively. Consider Z = X + Y, and its probability mass function pz(z). (a) (5 pts) Prove that Pz(z) = Σpx(x)py (z − x). (Hint: Show that the event "Z Yzx" for x = 0,..., z.) = Z = x=0 z" is the union of the disjoint events "X = x AND (b) (5 pts) Use part (a) to prove that if X~ Binom(n, a), Y~ Binom(m, a), then Z~ Binom(n + m, a). x=0 (Hint: First prove that Σ²_o (") (m₂) = (n+m). This can be proved by induction, or by considering how many ways one can choose z items among n + m items, where the items are divided into two groups of sizes n and m. Here we assume = 0 if x <0 or x > n.)

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