I think it is pretty easy to find an unbiased estimator for a regular distribution, whether it be Poisson or Gamma or something else. For example, the unbiased estimator (Poisson from a random sample of size $n$) for $\lambda$ would be $\overline{Y}$.
However, now suppose you have a function to find the number of failings of a computer system, and it is $C=2Y+Y^2$.
We can easily see that $E(C)=E(2Y + Y^2) = 3\lambda + \lambda^2$. But now if we want to find a function of $\overline{Y}$ that is an unbiased estimator of $E(C)$, how would you go about that?
I would think that you need to find $\hat{\theta}$, such that $E(\hat{\theta})=3\lambda + \lambda^2$, but I am a little confused as to whether or not that is accurate.
Any help on this problem would be greatly appreciated! Thank you very much.
Recall that for every random variable $X$ Poisson with parameter $\lambda$ one has $E(X)=\lambda$ and $E(X^2)=\lambda^2+\lambda$ hence $\mathrm{var}(X)=\lambda$. Thus, if $(Y_k)$ is i.i.d. Poisson $\lambda$ and $\bar Y=\frac1n\sum\limits_{k=1}^nY_k$ then $\bar Y$ is an unbiased estimator of $\lambda$ since $E(\bar Y)=\lambda$.
Likewise, $\mathrm{var}(\bar Y)=\frac1{n}\mathrm{var}(Y_1)=\frac1n\lambda$ hence $E(\bar Y^2)=\frac1n\lambda+\lambda^2$. Solving for $\theta=3\lambda+\lambda^2$ yields $\theta=E(\bar Y^2)+(3-\frac1n)E(\bar Y)$ hence an unbiased estimator of $\theta$ is $$ \Theta=\bar Y^2+\left(3-\frac1n\right)\bar Y. $$