Let $X_1,X_2,\dots,X_n$ be a random sample from distribution with pmf $p(x;θ)=\theta^x(1−θ),x=0,1,2,…;0$ elsewhere, $0≤θ≤1$. I know that $Y_1=\sum_1^n X_i$ is a sufficient statistic for $θ$ because this pmf itself comes from family of exponential distribution. The expectation for sufficient statistic is $$E[Y_1]=n\frac{1−θ}{θ}$$ How do I find a function of Y1 that is an unbiased estimator for $θ$? I tried algebraically solving it and the function is $\varphi(Y_1)=\frac{n}{Y_1+n}=\frac{1}{\frac{Y_1}{n}+1}$. Is this correct? I'm not sure how to find the $\mathbb{E}[\varphi(Y_1)]$ though.

  • $\begingroup$ I replaced the probability-theory tag by probability. Please avail yourself of the tag summaries when choosing tags. $\endgroup$ – joriki Mar 26 at 3:57
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    $\begingroup$ Your expectation is the wrong way around; it should be $\frac{n\theta}{1-\theta}$. $\endgroup$ – joriki Mar 26 at 3:58