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Math Help - find expectation of this r.v.

  1. #1
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    find expectation of this r.v.

    toss balls one at a time into n bins, each ball will always land in one of the n bins.
    stop tossing once some bin end up with 2 balls. and the tosses are independent of each other

    X be the number of tosses needed.
    (so X is between 2 and n+1)
    find E(X)

    i find linearity hard to apply here.
    and the naive definition resulted in a messy sum, that i cannot reduce to simple form.
    is this a well know distribution somewhere?

    thanks for any insights
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: find expectation of this r.v.

    Quote Originally Posted by pv3633 View Post
    toss balls one at a time into n bins, each ball will always land in one of the n bins.
    stop tossing once some bin end up with 2 balls. and the tosses are independent of each other

    X be the number of tosses needed.
    (so X is between 2 and n+1)
    find E(X)

    i find linearity hard to apply here.
    and the naive definition resulted in a messy sum, that i cannot reduce to simple form.
    is this a well know distribution somewhere?

    thanks for any insights
    Hi pv3633!

    This is not a well known distribution as far as I know.

    The formula for EX is:

    EX=\sum_{k=2}^{n+1} k \cdot \frac {n!(k-1)!} {(n-k+1)!n^k}

    Wolfram|Alpha could not solve this (within its timeout).
    But I found that a close numerical approximation is:

    EX \approx \frac 5 4 \sqrt n + \frac 3 4
    Thanks from pv3633
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  3. #3
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    Re: find expectation of this r.v.

    thank you.
    that's the sum i got also.

    how to get the approximation btw?
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: find expectation of this r.v.

    Quote Originally Posted by pv3633 View Post
    thank you.
    that's the sum i got also.

    how to get the approximation btw?
    I calculated a couple of values with Wolfram|Alpha:
    Code:
    n	EX
    1	2
    10	4.6
    100	13.2
    1000	40.3
    10000	125.66
    Then I made a log-log-plot in excel which showed a straight line.
    I used excel's solver to find the coefficients.

    The resulting approximation is:
    Code:
    n	EX	Approx
    1	2	2
    10	4.6	4.702847075
    100	13.2	13.25
    1000	40.3	40.27847075
    10000	125.66	125.75
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  5. #5
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    Re: find expectation of this r.v.

    thank you
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