Le $X_1, \ldots,X_n$ be samples of $\mathcal{N}_1(\mu,1)$ and $Y_n$'s are independent of $X_1, \ldots,X_n$ s.t. $$\Pr(Y_n=1)=n^{-2}=1−\Pr(Y_n=0)$$
- Let $L$ be MLE of $\mu$ by $X_1,\ldots,X_n$, and put $Z_n=(1−Y_n)L+\sqrt{n} Y_n$. What is $$\lim_{n \to \infty}\operatorname{Var}(\sqrt{n}(Z_n−\mu))?$$
So far I calculated the following:
\begin{align} \operatorname{Var}(\sqrt{n}((1-Y_n)L+\sqrt{n}Y_n-\mu)) & = \operatorname{Var}(\sqrt{n}(L-\mu) + nY_n -\sqrt{n}Y_nL) \\ &= \operatorname{Var}(\sqrt{n}(L-\mu)) + n^2\operatorname{Var}(Y_n) - \operatorname{Var}(\sqrt{n}L)Var(Y_n) \end{align}
Is that mathematically correct? Furthermore, I calculated $\operatorname{Var}(Y_n)$ which should be $$\operatorname{Var}(Y_n) = \mathbb{E}[Y_n^2]- \mathbb{E}[Y_N]^2 = \dfrac{n^2-1}{n^4}$$
So can I say the limit equals to $$\lim_{n \to \infty}\operatorname{Var}(\sqrt{n}(Z_n−\mu)) = 1 + 1 = 2?$$
Thank you very much in advance for helping me.