Let X Exp() (exponential random variable with parameter > 0) and Y Geom(p) (geometric random variable with parameter p (0, 1)). Show that both X and Y satisfy the "memoryless" property for random variables, that is P(Z > s|Z > t) = P(Z > s t).

That is the probability that the random variable will take values greater than s given the information that it has already taken values greater than t is the same as the probability that the random variable takes values greater than s t.

That is, we only need to take into account what has happened from time t to time s (anything that happened before time t is irrelevant to the probability calculation above).

Hint: You will need to calculate the cdfs of both X and Y . Note that P(Z > s|Z > t) = P(Z > s and Z > t) P(Z > t) .