Let $X_1, X_2,..., X_n$ be iid random variables with probability density function $$f(k|\theta) = \begin{cases} \theta, & \text{if $k=0$} \\ \theta(1-\theta), & \text{if $k=1$}\ \ \ \ \ \ \ \ \ \ \ \text{with 0<$\theta$<1}\\ (1-\theta)^2, & \text{if $k=2$} \end{cases}$$ Let $n_k$ be the number of observations for $k$ ($k \in (0,1,2)$). We observe two estimators $T_1$, $T_2$ for $\theta$ defined as follows:
$T_1 = \frac{n_0}{n}$. $T_2$ is the maximum likelihood estimator (MLE) for $\theta$.
I've calculated the MLE, which is: $$\hat\theta = \frac{n_0+n_1}{n_0 + 2(n_1 + n_2)}$$
Now I'm trying to find see if $T_1, T_2$ are unbiased and/or consistent. Thus I need to find $\mathbb{E}(T_1)$, $\mathbb{E}(T_2)$, Var($T_1$) and Var($T_2$).
My thoughts:
$$\mathbb{E}(T_1) = \mathbb{E}\left(\frac{n_0}{n}\right) = \frac{n \theta}{n}=\theta$$ So $T_1$ is an unbiased estimator for $\theta$. However I'm stuck at calculating the variance. I know that Var(X) = $\mathbb{E}(X^2)$-$\mathbb{E}(X)^2$. But I just don't see it now. What am I missing?
And how can I calculate $\mathbb{E}(T_2)$ = $\mathbb{E}\left(\frac{n_0+n_1}{n_0+2(n_1+n_2)}\right)$? Which $n$ am I supposed to use? And why?