Exponential Distribution and Waiting Time

[gajohnson]

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?

 

[Ray Vickson]

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?

Yes, but even more than that is true: you can even say what is the distribution of W, and the question asks you to do that.

 

[gajohnson]

Yes, but even more than that is true: you can even say what is the distribution of W, and the question asks you to do that.

Yes, of course. But it works out pretty simply that the distribution of W is the same as the distribution of Z. Thanks for confirming this for me. Sometimes something seems so simple that I question it--an instance of trying to disentangle actual mathematical results from the intentions of those writing the exercises.

 

[Ray Vickson]

Yes, of course. But it works out pretty simply that the distribution of W is the same as the distribution of Z. Thanks for confirming this for me. Sometimes something seems so simple that I question it--an instance of trying to disentangle actual mathematical results from the intentions of those writing the exercises.

You may not be aware of it, but the memoryless property of the exponential---in all its glory and detail--- is perhaps one of the most useful facts in probability. It is used everywhere, especially in queueing theory, reliability modelling, etc. This problem is introducing you to one of the most important topics in the subject, so is not just a "busy work" homework problem.

 

[gajohnson]

You may not be aware of it, but the memoryless property of the exponential---in all its glory and detail--- is perhaps one of the most useful facts in probability. It is used everywhere, especially in queueing theory, reliability modelling, etc. This problem is introducing you to one of the most important topics in the subject, so is not just a "busy work" homework problem.

I am indeed aware of the importance of the memoryless property of the exponential (although likely not to the extent that you are), and this is precisely why it seemed to me that there was a trick involved.

I don't know quite how I gave off the impression that I thought the memoryless property was itself trivial, I only meant to say that I was suspicious that the solution was a trivial application of that property. However, if I have accidentally implied differently, you defended its honor nobly.

 

[Ray Vickson]

I am indeed aware of the importance of the memoryless property of the exponential (although likely not to the extent that you are), and this is precisely why it seemed to me that there was a trick involved.

I don't know quite how I gave off the impression that I thought the memoryless property was itself trivial, I only meant to say that I was suspicious that the solution was a trivial application of that property. However, if I have accidentally implied differently, you defended its honor nobly.

Well, the magic is that it is "easy", but in a way very deep. The math is trivial but the consequences are far from trivial.

 

[gajohnson]

I look forward to discovering more about this wonderful thing. Thanks for your insight!