Let's say we have a random variable $X$ defined the following way:
$$X: \begin{pmatrix} 1 & 2 & 3 \\ \frac{1}{2}-p & p & \frac{1}{2}-p \end{pmatrix}$$
a) find all possible values for $p$
b) estimate the parameter $p$ when we have the following sample $(1, 3, 2, 1, 1, 3, 1)$
c) if sample has $n$ elements then find estimate of a $p!$ where we have 1 appearing $n_1$ times, number 2 appearing $n_2$ times and 3 appearing $n-n_1-n_2$ times
What i have so far is the following:
a) we have that $|\frac{1}{2}-p| <1$ and $|p|<1$
so $p \in (0,\frac{1}{2})$
b) this one is the big deal for me, how am i supposed to estimate $p$ based on the sample i have, i need to mention that this is the first time i am trying to solve this kind of problem, and i have no idea what i am supposed to do here, any help appreciated!