Let X be a discrete random variable whose PMF is
$\qquad p(0) = \frac{2\theta}{3}, \quad p(1) = \frac{\theta}{3}, \quad p(2) = \frac{2(1-\theta)}{3}, \quad p(3) = \frac{1-\theta}{3}$.
Suppose that $X_1, \dots, X_n$ are iid samples of X ($n > 0)$.
(a) Find the MME for $\theta$. [10 marks]
(b) Is the MME unbiased? Is it asymptotically unbiased? [10 marks]
(c) Find the MLE for $\theta$. [10 marks]
(d) Is the MLE unbiased? Is it asymptotically unbiased? [10 marks]