- #1

gajohnson

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## Homework Statement

Suppose that the waiting time for the CTA Campus bus at the Reynolds Club stop is a continuous random variable Z (in hours) with an exponential distribution, with density f(z) = 6e–6z for z ≥ 0; f(z) = 0 for z < 0.

(a) What is the expected waiting time in minutes (the expected value of Z)?

(b) Suppose you have been waiting exactly ½ hour. What is the expected additional waiting time

E(W), where W = Z – ½ ? [Hint: For a > ½ , what is the conditional probability Z > a, given Z > ½ ? What is the conditional probability Z < a, given Z > ½ ? What is the conditional probability W < b=a–0.5, given Z > ½ ? What is the conditional density of W?]

## Homework Equations

## The Attempt at a Solution

Part (a) is easy, simply doing the expected value calculation and coming away with E(Z)=1/6.

Isn't the answer the part (b) 1/6 as well due to the memoryless property of the exponential distribution? Or am I misunderstanding the question and/or the memoryless property? If you've already been waiting 1/2 hour, the expected additional waiting time is the same as the expected waiting time at time 0, is it not?