Unbiased sufficient statistic for $1/p$ of geometric distribution

Question

Suppose that a random variable $X$ has the geometric distribution with unknown parameter $p$, where the geometric probability mass function is: $$ f(x;p)= p(1-p)^x,\qquad x=0,1,2,\ldots ; \quad 0<p<1$$ Find a sufficient statistic $T(X)$ that will be an unbiased estimator of $1/p$.

Now I know the population mean is $\frac{1-p}{p}$ and the sufficient statistic for p is a function of $\sum_{i=1}^{n}X_i$. But I am unsure on how to proceed any help greatly appreciated!

Answer 1

Assuming $X_1,X_2,\ldots,X_n$ are i.i.d with pmf $f$.

You have $E_p(X_1+1)=\frac1p$ for all $p\in(0,1)$, so

$$E_p\left[\frac1n\sum_{i=1}^n (X_i+1)\right]=E_p\left[\frac1n\sum\limits_{i=1}^n X_i+1\right]=\frac1p\quad,\,\forall\,p\in(0,1)$$