Find an unbiased estimator for Poisson distribution

Question

I am starting studying statistics and I have a task to find unestimated bias for Poisson distribution.
The task sounds like this: "Find unbiased estimator $\lambda_{s} = \lambda^{3}$ using sample $X_{1}...X_{n}$ from $Pois(\lambda)$ distribution".

I know that $\lambda$ is variance in Poisson distribution and that variance unbiased estimator is $$\bar{x}=\frac{1}{N}\sum_{i=1}^{N}X_{i}$$ But I see I need an answer correlated with $\lambda_{s}$, and I don't know how to use Poisson distribution formula to get my $X_{1}...X_{n}$ and then unbiased estimator.

Do you have any ideas?

P.S. Sorry, right problem statement is with UNbiased estimator...

Answer 1

First of all, notice that if $X\sim \text{Poisson}(\lambda)$ then $$E(X)=\lambda$$ $$E(X^2)=\lambda^2 + \lambda$$ $$E(X^3)=\lambda^3 +3\lambda^2 +\lambda$$ This implies $$E(X^3-3X^2+2X)=\lambda^3$$ That being said, define $$Y_{N}=\frac{1}{N}\sum_{i=1}^{N}\Big[X_i^3-3X_i^2+2X_i\Big]$$ Using linearity of expectation we have $E(Y_{N})=\lambda^3$ so $Y_{N}$ is an unbiased estimator for $\lambda^3$.