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Math Help - Help with this integral

  1. #1
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    Help with this integral

    Define, for x \geq y \geq 0
     \Lambda(x,y)=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(  x-y+1)}
    Then proof for all n>0
    \int^n_{0}\Lambda(n,x)dx=2^n
    And also prove for 0<p<1
    \int^ \infty_{-\infty}p^x(1-p)^{n-x}\Lambda(n,x)dx=1
    I once asked someone this question and they said it looks like the Beta Function, thus maybe that will help.

    If you are curious how I came to such a conjecture is because I was trying to describe a mathematical formula for the Normal Distribution and these two:
    \Lambda(n,x)
    and
    p^x(1-p)^{n-x}\Lambda(n,x) where p is the probability.
    Graph curves which look like the Normal Distribution, furthermore the second one is a density curve this is why its Area is one.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker
    Define, for x \geq y \geq 0
     \Lambda(x,y)=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(  x-y+1)}
    Then proof for all n>0
    \int^n_{0}\Lambda(n,x)dx=2^n
    If this were true then:

    I(1)=\int_0^1 \frac{\Gamma(2)}{\Gamma (x+1) \Gamma(2-x)} dx =2.

    But it seems to me that I(1)\approx 1.2.

    RonL
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    CaptainBlack, Graph it and take its integral after that. Maybe, I am making a mistake in the way I posed the problem. Or perhaps, I made a mistake in the domain of the function. Anyways, would you confirm that these two functions look like the normal distribution curve?

    I would check it but I "borrowed" someone else's computer and it does not have a graphing program.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker
    CaptainBlack, Graph it and take its integral after that. Maybe, I am making a mistake in the way I posed the problem. Or perhaps, I made a mistake in the domain of the function. Anyways, would you confirm that these two functions look like the normal distribution curve?

    I would check it but I "borrowed" someone else's computer and it does not have a graphing program.
    IIRC

    <br />
p^x(1-p)^{n-x}\Lambda(n,x)<br />

    with x and n positive integers is the probability of x successes in n
    trials, where p is the probability of success in a single trial.. This is the binomial
    distribution and is well know to approximate the normal distribution (in a
    rather peculiar sense of approximate) for large n

    RonL
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