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Math Help - unbiased estimators

  1. #1
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    unbiased estimators

    suppose that a random variable X has a geometric distribution (pq^x) for which the parameter p is unknown (0<p<1). Find a statistic that will be an unbiased estimator of 1/p

    Any help on this would be greatly appreciated, thank you so much
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  2. #2
    MHF Contributor matheagle's Avatar
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    Do you have a sample or just one observation?
    You do realize E(X)=1/p depending on how you define your distribution
    http://en.wikipedia.org/wiki/Geometric_distribution
    IN your case E(X)=(1/p)-1
    So, in that setting X+1 is the answer
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  3. #3
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    Is there a way to prove X+1 is an unbiased estimator of 1/p? We usually write out proofs which I have not gotten the hang of yet
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  4. #4
    MHF Contributor matheagle's Avatar
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    IF you are using P(X=x)=pq^x for x=0,1,2,3...

    Then E(X)=(1/p)-1, do you need a proof of that?

    THEN E(X+1)=E(X)+1=(1/p)-1+1=1/p.
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  5. #5
    MHF Contributor matheagle's Avatar
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    OR for P(X=x)=pq^{x-1} for x=1,2,3...

    then E(X)=1/p. IN that case X is the answer
    But do you have a sample X_1, X_2,....? or just one X?
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  6. #6
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    i think it is a sample
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  7. #7
    MHF Contributor matheagle's Avatar
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    Then use \bar X+1
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