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Exponential Distribution and Waiting Time

  1. Oct 16, 2013 #1
    1. The problem statement, all variables and given/known data

    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?]


    2. Relevant equations



    3. 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?
     
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  3. Oct 17, 2013 #2

    Ray Vickson

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    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.
     
  4. Oct 17, 2013 #3
    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.
     
  5. Oct 17, 2013 #4

    Ray Vickson

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    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.
     
  6. Oct 17, 2013 #5
    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.
     
  7. Oct 17, 2013 #6

    Ray Vickson

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    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.
     
  8. Oct 17, 2013 #7
    I look forward to discovering more about this wonderful thing. Thanks for your insight!
     
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