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Let X Exp() (exponential random variable with parameter > 0) and Y Geom(p) (geometeric 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 great 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)

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Let X Exp exponential random variable with parameter g.pdf

  • 1. Let X Exp() (exponential random variable with parameter > 0) and Y Geom(p) (geometeric 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 great 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)