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Probability theory: a regenerative process

  1. Oct 13, 2008 #1
    1. The problem statement, all variables and given/known data
    [​IMG]


    3. The attempt at a solution
    First of all I'm trying to find the expected time of a cycle. In a cycle two things can happen:

    1) the car lives long enough to reach A with probability 1-F(A)
    2) the car fails to reach age A with probability F(A)

    Knowing this how can you compute the liftime of a cycle? I thought of something like:

    E(T) =A*(1-F(A)) + F(A) * ? (I don't know what to fill in the question mark)
     
  2. jcsd
  3. Oct 13, 2008 #2

    EnumaElish

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    E(T) is the expected value of T ~ F conditional on T < A. But you can directly solve for the expected resale price:

    Let event 1 be {the car fails before A}. If event 1 occurs then resale value = 0 (do you see this?)
    Let event 2 be {the car lives to A} (that is, it lives to see A, and possibly more). If event 2 occurs then resale value = R(A).

    Suppose the probability of event 1 was G. Convince yourself that the prob. of event 2 must be 1 - G.

    Then E(resale value) = G * 0 + (1-G) * R(A).

    What you should think about is how to find G.
     
  4. Oct 13, 2008 #3
    So the expected life of a cycle is: [tex] E[T|T<A] = \int_0^{A} x f(x)\ \mbox{d}x[/tex]

    Yes It's stated in the exercise.

    That's clear.

    I presume that G must be F(A). But the question asks for the costs so how do I incorporate that in the resale value given that there are two different outcomes?
     
  5. Oct 13, 2008 #4

    EnumaElish

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    Once you figure the expected resale value, you can use it to figure the expected net cost, defined as cash expenses - expected resale value.
     
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