Let k and n be non-negative integers, and 0 < p < 1. A random variable...

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Let k and n be non-negative integers, and 0 < p < 1. A random variable X has a geometric distribution if its pmf is given by px (k) = (1 - p)pk. Define the residual lifetime distribution function as, lx (k, n) := P(X ≥n + k|X ≥ n). P(X k) independent of n, i.e., the geometric distribution satisfies the (a) Show that Ix (k,n) memoryless property. (b) Assume that Y≥ 0 is any other discrete integer-valued distribution which exhibits memoryless property, i.e., ly (k, n) = P(Y ≥ k). Show that ly (k, n) has to be of the form ak for some 0 < a < 1. (c) Using (b), show that if Y satisfies the memoryless property, then it has a geometric distribution. Let k and n be non-negative integers, and 0 < p < 1. A random variable X has a geometric distribution if its pmf is given by px (k) = (1 - p)pk. Define the residual lifetime distribution function as, lx (k, n) := P(X ≥n + k|X ≥ n). P(X k) independent of n, i.e., the geometric distribution satisfies the (a) Show that Ix (k,n) memoryless property. (b) Assume that Y≥ 0 is any other discrete integer-valued distribution which exhibits memoryless property, i.e., ly (k, n) = P(Y ≥ k). Show that ly (k, n) has to be of the form ak for some 0 < a < 1. (c) Using (b), show that if Y satisfies the memoryless property, then it has a geometric distribution.

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