# Minimum Variance Unbiased Estimator for Exponential Family of Distribution [duplicate]

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.

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