Given a Poisson distribution I want to figure out whether $d:(x_1,...,x_n) \mapsto x_1^2$ and $d':(x_1,...,x_n) \mapsto x_1x_2$ are unbiased estimations for $\lambda^2$ ?
I mean it would sound reasonable if they were, cause the expected value for every $x_1$ is $\lambda$ itself, but since the product of expected values is not necessarily the expected value of a product, this does not has to be true. So how does this work here?
Let us assume that $X_1$ and $X_2$ are independent. Then $E(X_1X_2)=E(X_1)E(X_2)=\lambda^2$.
Now we deal with $X_1^2$. It is a standard fact about the Poisson with parameter $\lambda$ that it has variance $\lambda$. It follows that $E(X_1^2)=\lambda+\lambda^2$, so $X_1^2$ is not an unbiased estimator of $\lambda^2$.
Remark: If the "standard fact" cannot be used, one needs to find $E(X_1^2)$ by evaluating the sum $$\sum_{1}^\infty k^2e^{-\lambda}\frac{\lambda^k}{k!}.$$ This can be done in various ways. Note for example that $\frac{k^2}{k!}=\frac{1}{(k-1)!}$.