# MAT8886 the Dirichlet distribution

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In the course, still introducing some concept of dependent distributions, we will talk about the Dirichlet distribution (which is a distribution over the simplex of ). Let denote the Gamma distribution with density (on )

Let denote independent random variables, with . Then where

has a Dirichlet distribution with parameter . Note that has a distribution in the simplex of ,

and has density

We will write .

or and , below

and finally, below,

Note that marginal distributions are also Dirichlet, in the sense that if

> library(MCMCpack) > alpha=c(2,2,5) > x=seq(0,1,by=.05) > vx=rep(x,length(x)) > vy=rep(x,each=length(x)) > vz=1-x-vy > V=cbind(vx,vy,vz) > D=ddirichlet(V, alpha) > persp(x,x,matrix(D,length(x),length(x))(to plot the density, as figures above). Note that we will come back on that distribution later on so-called Liouville copulas (see also Gupta & Richards (1986)).

To

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