Question 2. Let X be a discrete random variable with probability mass function $f(x; \theta)$ for
a scalar parameter $\theta$. We assume that $\theta$ can only take on two values, $\theta = \theta_1$ or $\theta = \theta_2$.
The probability mass function of X, $f(x; \theta)$, is given by
$f(x; \theta_1) = \begin{cases} 1/3, & \text{if } x = -1 \\ 2/3, & \text{if } x = 1 \end{cases}$
and
$f(x; \theta_2) = \begin{cases} 2/3, & \text{if } x = -1 \\ 1/3, & \text{if } x = 1. \end{cases}$
Suppose we observe a single value of X, $x = -1$. What is the maximum likelihood estimate
of $\theta$? (5 marks).
