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Math Help - Maximum likelihood estimator of the ratio

  1. #1
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    Maximum likelihood estimator of the ratio

    Please help me is this statistical problem.

    An urn contains black and white balls. A sample of size n is drawn with replacement. What is the maximum-likelihood estimator of the ratio R of black to white balls in the urn?
    Suppose that one draws balls one by one with replacement until a black appears.
    Let X be the number of draws required (not counting the last draw).
    This operation is repeated n times to obtain a sample X1, X2, ..., Xn.
    What is the maximum-likelihood estimator of R on the basis of this sample.
    Last edited by imjae; December 9th 2010 at 09:30 PM. Reason: title is not appropriate
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by imjae View Post
    Please help me is this statistical problem.

    An urn contains black and white balls. A sample of size n is drawn with replacement. What is the maximum-likelihood estimator of the ratio R of black to white balls in the urn?
    Suppose that one draws balls one by one with replacement until a black appears.
    Let X be the number of draws required (not counting the last draw).
    This operation is repeated n times to obtain a sample X1, X2, ..., Xn.
    What is the maximum-likelihood estimator of R on the basis of this sample.
    The X_i s are independent identically distributed RVs with:

    p(X_i=k_i)=(1-\rho)^{k_i-1}\rho

    where \rho=R/(1+R)

    So now write out the likelihood of outcome k-1, k_2, .., k_n ..
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    The X_i s are independent identically distributed RVs with:

    p(X_i=k_i)=(1-\rho)^{k_i-1}\rho

    where \rho=R/(1+R)

    So now write out the likelihood of outcome k-1, k_2, .., k_n ..

    Thank you for your reply.
    May I know why p = R/(1+R)?
    Can we let R be equal to b/w, where b=no. of black balls and w=no. of white balls?
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