Question: 20 Let X be a geometric random variable with parameter p. and n and m be nonnegative integers. (a) Prove that P(X>n+m | X>m) =

20 Let X be a geometric random variable with parameter p. and n and m be nonnegative integers.

(a) Prove that P(X>n+m | X>m) = P(X >n). What is the meaning of this property in terms of Bernoulli trials? This is called the memoryless property of geometric random variables.

(b) For what values of n is P(X =n) maximum?

(c) What is the probability that X is even?

(d) Show that the geometric is the only distribution on the positive integers with the memoryless property.

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