Let $X_1,X_2,...,X_{100}$ be observations from a $X_i \sim \operatorname{Poisson}(\lambda)$ for $i =1,...,100$, where $\lambda > 0 $ is an unknown parameter. The following estimates for $\lambda$ are given by: $$ \widehat{\Theta}_{50} = \frac{1}{50}(X_1+X_2+...+X_{50}) $$ and $$ \widehat{\Theta}_{100} = \frac{1}{100}(X_1+X_2+...+X_{100}) $$ Determine which estimate is best by using mathematical notation.
First I have just written that more observations are better in general but I have also found the MSE for both estimates:
$$ \begin{split} MSE &= E[(\widehat{\Theta}-\theta)^2] \\ &= \operatorname{Var}(\,\overline{\!X}-\theta)+(E[\widehat{\Theta}-\theta])^2 \\ &= \operatorname{Var}(\,\overline{\!X}) \\ &= \frac{1}{n^2} \operatorname{Var}(X_i) \\ &= \frac{1}{n^2} \sum_{i=1}^n \operatorname{Var}(x) \\ &= \frac{\lambda}{n} \end{split} $$ Thus resulting in $MSE(\widehat{\Theta}_{50}) = \frac{\lambda}{50} > MSE(\widehat{\Theta}_{100}) = \frac{\lambda}{100}$ whereby I would conclude that the smaller MSE the better and would argue that 50 observartions < 100 observations.
Am I missing any argumentation? This is an earlier exam question from my course which I am studying for.
Thanks.