I am unclear on the how to write the likelihood function in order to find the maximum likelihood estimate of $\theta$ from the following probability mass function (PMF), if I have 5 independent observations $(3,0,2,1,3)$.
$$P(X\vert\theta)=\begin{cases}{\frac{2\theta}{3}}& x=0 \\ \frac{\theta}{3}& x=1\\ \frac{1-\theta}{3} &x=2 \\ \frac{2(1-\theta)}{3} & x=3 \end{cases}$$
Now it is my understanding that the likelihood function is the product (in this case five times) of the PMF. Since each $i$ observation is different I have to write the corresponding function and multiply it by the next in the following way:
$$L(\theta|X)=\frac{2(1-\theta)}{3} \cdot \frac{2\theta}{3} \cdot \frac{1-\theta}{3} \cdot \frac{\theta}{3} \cdot \frac{2(1-\theta)}{3}$$
This seems right but I have the doubt (unfounded and irrational maybe?) that it should also include the realized values, like this:
$$L(\theta|X)=(3)\frac{2(1-\theta)}{3} \cdot (0)\frac{2\theta}{3} \cdot (2)\frac{1-\theta}{3} \cdot (1)\frac{\theta}{3} \cdot (3)\frac{2(1-\theta)}{3}$$
I hope I'm correct in using the first likelihood function, but if it is the second one could you please explain why?