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Math Help - Poisson to Exponential through integration

  1. #1
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    Poisson to Exponential through integration

    Alright, Ive asked many of my fellow students (who seem to be as lost as I am) and a few of my professors (none of whom really help in explaining it despite their good intentions), so hopefully you guys can help me

    My professor said we're going to have to solve a problem that will take a Poisson distribution to Exponential. I understand that if you have a Poisson distribution where you only take 1 value, it can be simplified to \lambda e^{-\lambda x}. Similarly, the exponential equation is (1/\mu)e^{-(1/\mu)x)}.

    First, does this mean that \lambda = (1/\mu) ?

    Second, he said that we were going to have to use integration to solve a problem. The example that he gave us where we'll have to take from poisson to exponential is f(x)=cx^2 for say 2<x<5. How would I do this through integration?
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  2. #2
    MHF Contributor matheagle's Avatar
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    I don't understand the question.
    How do you 'take' a discete random random variable into a continous one?
    The waiting times in a Poisson process are gamma random variables, is that what you're asking?
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  3. #3
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    I don't exactly know myself what he is going for either.

    He gave us many examples of integration such as

    \int 5e^{-1/10000x}dx
    -(1/10000)e^{-1/10000x}
    Therefore the final answer is -50000e^{-1/10000} + C

    That somehow is supposed to relate to going from Poisson to Exponential. He said, on the test, we'd get something like  f(x)=CX^2 and we'll be expected to integrate it from Poisson to Exponential.
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  4. #4
    MHF Contributor matheagle's Avatar
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    I didn't understand the question nor this integration.
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  5. #5
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    wow we belong in same class..

    First, does this mean that \lambda = (1/\mu) ?
    YES.

    Second, he said that we were going to have to use integration to solve a problem. The example that he gave us where we'll have to take from poisson to exponential is f(x)=cx^2 for say 2<x<5. How would I do this through integration?[/quote]
    This also confused me also since it has nothing to do with poisson or exponential distribution. The only realistic reason i see is that
    he tried to explain how the exponential function could be integrated to exponential cumulative probability
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  6. #6
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    I'm curious since in Poisson process the value t should go infinitely to 0
    using poisson function that would make  f(x)=<br />
\ e^{-\lambda t}<br />
    where would <br />
\lambda<br />
come from?
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  7. #7
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    Haha, no way? Really? You're in Mr Park's class? Man, feel free to introduce yourself next week. Im the big, tall white guy (IE the only one in class). I'll be looking out for you.

    From his explanation, I pretty much got the lambda equals 1/mu. I understand the relationship between the two equations and I agree with you that the integration doesn't make too much sense. If it makes you feel any better, Ive talked to 4 different students in that class (1 of whom has talked to the TA about it) and none of them understand how the integration relates Poisson and the exponential.


    I'm curious since in Poisson process the value t should go infinitely to 0
    using poisson function that would make
    where would come from?

    As for this, I'll let someone else answer it. Can someone please help here?
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  8. #8
    MHF Contributor matheagle's Avatar
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    Maybe this is what you're looking for...
    http://imperator.usc.edu/~bruck/clas...xponential.pdf
    As I said yesterday, the waiting times in a Poisson Process are exponential.
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