We have $n$ independent observations, $\{Y_1,...,Y_n\}\overset{iid}\sim\mathcal{Poissson}(\mu)$. Consider the estimator $\tilde\mu = \bar Y$. I have to show that $\tilde\mu$ is unbiased and find the variance. Did I do this correctly?
$$\begin{align}\mathbb{Bias}(\tilde\mu) &= \mathbb E\tilde\mu-\mu\\ &= \mathbb E(\tfrac 1n\sum_{i=1}^n\mathbb E(Y_i))-\mu\\ &= \tfrac 1n\sum_{i=1}^n\mathbb E(Y_i)-\mu\\&=\mu-\mu\\&=0 \\ \\\therefore\quad \mathbb{Var}(\mu) &=\mu&&\text{Since }\{Y_1,...,Y_n\}\overset{iid}\sim\mathcal{Poissson}(\mu)\end{align}$$
The variance is incorrect. Note that $$ \text{Var}(\bar{Y})=\frac{1}{n^2}\sum \text{Var}(Y_i)=\frac{1}{n^2}\times n\mu=\frac{\mu}{n} $$ since the $Y_i$ are independent.