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Thread: Expected value: Suppose people randomly pick a number between 0 and 1....

  1. #11
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    Quote Originally Posted by Jomo View Post
    Hi,
    Suppose people randomly pick a number between 0 and 1. What would be the expected number of people you would have to ask for numbers if you want the sum to be greater than 1 for the 1st time?
    Steve
    after toying with this for a while I think i've got it.

    the probability it takes [tex]n[/tex] uniform[0,1] addends to sum to greater than 1 is given by

    [tex]\Large \displaystyle \int_0^1 \int_0^{1-x_1}\int_0^{1-x_1-x_2} \dots \int_0^{1-\sum \limits_{k=1}^{n-2}x_k}\int_{1-\sum \limits_{k=1}^{n-1}~x_k}^1~1~dx_n~dx_{n-1}~\dots dx_2~dx_1[/tex]

    This generates numbers in agreement with what I saw in the sim.

    This is the integral over the probability mass where the first (n-1) variables sum to less than one but the sum over all n sums to greater than 1.

    [tex]p[2]=\dfrac 1 2\\p[3]=\dfrac 1 3\\p[4] = \dfrac 1 8\\p[5]=\dfrac{1}{144} \\ p[k] = \dfrac{1}{k!+(k-1)!}[/tex]

    [tex]E[k]=e[/tex]
    Last edited by Romsek; 12-07-2018 at 04:10 AM.

  2. #12
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    Quote Originally Posted by Romsek View Post
    after toying with this for a while I think i've got it.

    the probability it takes [tex]n[/tex] uniform[0,1] addends to sum to greater than 1 is given by

    [tex]\Large \displaystyle \int_0^1 \int_0^{1-x_1}\int_0^{1-x_1-x_2} \dots \int_0^{1-\sum \limits_{k=1}^{n-2}x_k}\int_{1-\sum \limits_{k=1}^{n-1}~x_k}^1~1~dx_n~dx_{n-1}~\dots dx_2~dx_1[/tex]

    This generates numbers in agreement with what I saw in the sim.

    This is the integral over the probability mass where the first (n-1) variables sum to less than one but the sum over all n sums to greater than 1.

    [tex]p[2]=\dfrac 1 2\\p[3]=\dfrac 1 3\\p[4] = \dfrac 1 8\\p[5]=\dfrac{1}{144} \\ p[k] = \dfrac{1}{k!+(k-1)!}[/tex]

    [tex]E[k]=e[/tex]
    Yep, you got the correct result.
    A mathematician is a blind man in a dark room looking for a black cat which isnít there. - Charles R. Darwin

  3. #13
    Elite Member
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    Quote Originally Posted by Jomo View Post
    Hi,
    Suppose people randomly pick a number between 0 and 1.
    This is meaningless until you specify a probability distribution.

  4. #14
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    Quote Originally Posted by HallsofIvy View Post
    This is meaningless until you specify a probability distribution.
    Come on Halls, you're better than that. Someone says randomly between zero and one and nothing else it's uniform[0,1].

    If they come back and say, that's not what they meant then you can rail on them.

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