Under the name of mmcdara (unfortunately inaccessible since the major July 2025 Mapleprimes outage, and probably lost forever, God rest his soul.) I published two years ago a post about Multivariate Normal Distribution.

The current post continues in the same vein and presents the construction of a few new Multivariate Random Variables (MRV for short) named Multinomial (see for instance this recent question), Dirichlet, Categorical and related compound distributions.
I advice the interested readers to give a quick look to these names on Wikipedia (more specific references are given at the top of the wotksheet).

As I explained (in fact as my alter ego did) in Multivariate Normal Distribution, the Statistics package is limited to univariate random variabled  and thus implementing MRVs requires a little cunning.
Here is a list of a few problems you face:

• Whereas the expectation (sometimes named "mean") of a univariate random variable is a number or an expression, the expectation of a MRV is a vector (or a list, a n-uple, ...) of numbers or expressions.

So far, so good, except that the Mean attribute of Distribution can only be a scalar quantity. So if you want to assign a vector to Mean you have to code it some way and do something like Decode(Mean(My_MRV)) to get the expectation in a vector form.
 

• The Variance case is even more tricky because MRV variance are matrices.
   
• Beyond this some very useful attributes like ParentName and Parameters cannot be instanciated in the definition of user random variables (whether there are MRV or not), implying here again some bit of gymnastics to, if not reaslly instantiate these attributes, be able at least to retrieve them when needed.
   
• Finally, last but not least, the RandomSample is not appropriated to sample MRVs for reasons which are explained in the attached worksheet.

The file below contains more than 20 procedures enabling the definition of the studied MRVs, the decoding of the coded attributes, the visualization (which is not that immediate because the supports of the MRVs I foccus on are simplexes), the parameter estimations against empirical observations (frequentist and bayesian points of view), and so on.

Multinomial_Dirichlet_and_so_on.mw

Nevertheless, there is still a lot missing, but at some point I believe we need to decide that the work is over.