This is related to How to define an n-variate empirical distribution function probability for any n?
and
RandomVariate from 2-dimensional probability distribution
but I don't think neither questions (and their answers) answer the question below. (The first one constructs the problem from data and not from a priori specified probability weights, and hence they can use NProbability; and the second one seems like an extreme overkill for something like a simple discrete distribution where one doesn't really need to kick out random number generators).
I want to construct a multivariate discrete distribution so that I can use the full functionality of RandomVariate and things of that sort.
In 1-dimension, I can use EmpiricalDistribution. For instance, for a $X \sim Bernoulli(p)$ with $p = 1/2$, it is simply
gdist = EmpiricalDistribution[{0.5, 0.5} -> {0, 1}]
and from this, I can go on to compute mean and variances via Expectation, say
Expectation[ 2*x + 1, x \[Distributed] gdist]
as afforded by RandomVariate and all its friends.
Question: How does one do that for a multivariate discrete distribution (whether via EmpiricalDistribution or not)? That is, suppose we consider,
$$
(X,Y) =
\begin{equation}
\begin{cases}
(1,0), \quad \text{with probability } p_{10} \\
(0,1), \quad p_{01} \\
(0,0), \quad p_{00} \\
(1,1), \quad p_{11}.
\end{cases}
\end{equation},
$$
where of course $p_{10} + p_{01} + p_{00} + p_{11} = 1$ are the probability weights. How does one implement the above distribution, say labelled as gmultdist so that we can compute Expectation[ 2*x + 3*y, {x,y} \[Distributed] gmultdist]?
