Suppose that two random variables Y1 and Y2 are independent, that is, for all values of y1 and y2,P((Yı = y1) n (Y2 = y2)) = P(Y1 = y1)P(Y2 = 42)that is, the events (Y1 = y1) and (Y2 = y2) are independent. Suppose also that Y1 and Y2 have thesame Geometric distribution, that is Y1 - Geometric(p) and Ý, - Geometric(p). Define a thirdrandom variable Y as the sum of Y and Y2, that is, Y = Y1 +Y2. By considering the Theorem of Total Probability and the partitioning resulty-1(Y = y) = U ((Yı = t) n (Y2 = y – t))t=1for y = 2,3, ..., derive the pmf of Y.

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Suppose that two random variables Y1 and Y2 are independent, that is, for all values of y, and y2, P((Yı = y1) N (Y2 = y2)) = P(Y1 = y1)P(Y2 = y2) that is, the events (Yı = y1) and (Y2 = y2) are independent. Suppose also that Y1 and Y2 have the same Geometric distribution, that is Y ~ Geometric(p) and Ý, ~ Geometric(p). Define a third random variable Y as the sum of Y1 and Y2, that is, Y = Y1 +Y2.