In this question we’ll see how the memoryless property of the exponential distribution works.We’ll use X ∼ Exponential(λ =

4) for illustration.

a. Find an expression for P(X ≥ x) for any x > 0.

b. Now suppose that we condition on the event {X ≥ 2}.

Explain in words what P(X ≥ 2 + x | X ≥ 2) represents.

c. Using the definition of conditional probability, find an expression for P(X ≥ 2 + x | X ≥ 2). Simplify it as far as possible using the expression from part (a).

d. Based on parts

(a) and (c), you should have found that P(X ≥ 2 + x | X ≥ 2) = P(X ≥ x).

Explain in words why this is an example of memorylessness. What is it that has been forgotten?

e. Now suppose X ∼ Geometric(p). You may assume that P(X ≥ x) = (1 − p)

x

, for x = 0, 1, 2, . . . .

Repeat the calculation in

(c) to confirm that this produces another example of memorylessness.