Suppose that X0,X1,…,Xn are i.i.d. random variables that follow the Poisson distribution with mean λ. How can I prove that there is no unbiased estimator of the quantity 1λ?

**Answer**

Assume that g(X0,…,Xn) is an unbiased estimator of 1/λ, that is,

∑(x0,…,xn)∈Nn+10g(x0,…,xn)λ∑ni=0xi∏ni=0xi!e−(n+1)λ=1λ,∀λ>0.

Then multiplying by λe(n+1)λ and invoking the MacLaurin series of e(n+1)λ we can write the equality as

∑(x0,…,xn)∈Nn+10g(x0,…,xn)∏ni=0xi!λ1+∑ni=0xi=1+(n+1)λ+(n+1)2λ22+…,∀λ>0,

where we have an equality of two power series of which one has a constant term (the right-hand side) and the other doesn’t: a contradiction. Thus no unbiased estimator exists.

**Attribution***Source : Link , Question Author : billlee1231 , Answer Author : J. Virta*