Let X be a discrete random variable with values in N = {1, 2, . . .}. Prove

that X is geometric with parameter p = P(X = 1) if and only if the

memoryless property

P(X = n + m|X > n) = P(X = m)

holds.

Hint: To show that the memoryless property implies that X is geometric,

you need to prove that the p.m.f. of X has to be P(X = k) = p(1 p)k1.

For this, use P(X = k) = P(X = k + 1|X > 1) repeatedly.