I am having issues solving the follow problem from my textbook:
Suppose we have $x_1,...,x_n$ i.d.d. data from a r.v. $X$ with unknown distribution function $F_\theta$, for $\theta = (\theta_1,\theta_2)\in\Theta = [0,1] \times\mathbb R_+ \subset \mathbb R^2$. The r.v. $X$ is known to be of mixed type, with density
$$f_\theta(x) = \left\{ \begin{array}{c} \frac{1}{4}\theta_1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x = 1, \\ \frac{1}{4}(1 - \theta_1),\ \ \ \ \ \ \ x = 2, \\ \frac{3}{4}\frac{1}{\theta_2}e^{-(x-3)/\theta_2}, \ \ x\geq 3, \end{array} \right. $$
with respect to the measure $\mu(t) = \mu_0(t)\mathbb I\{t<3\} + \mu_1(t)\mathbb I\{t\geq3\}$, where $\mu_0$ is counting measure of the integers in $\mathbb R$ and $\mu_1$ is length measure on $\mathbb R$.
Derive the MLE of $\theta$.
When one deals with a variable of mixed type, I know that you consider the likelihood function $L(\theta) = \prod_{x_i < 3}f_\theta(x_i)\cdot\prod_{x_i \geq 3}f_\theta(x_i)$, but other that that I am not sure how to tackle this problem. So I am aware that I should begin by finding the MLE w.r.t. one of the parameters. But I don't see how one should do that when we don't know the actual values of the $x_i's$.
For example if none of the $x_i's = 1$, then $\theta_1 = 0$ is the maximiser, etc. This leads me to believe that there is something crucial I am missing here or is failing to consider on my own, and I haven't been able to find much help for this in the textbook that I'm using.