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In the course, since we are still introducing some concepts 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  wherehas a Dirichlet distribution with parameterNote that  has a distribution in the simplex of ,and has densityWe will write .The density for different values of  can be visualized below, e.g. , with some kind of symmetry,or  and , belowand finally, below, Note that marginal distributions are also Dirichlet, in the sense that ifthenif , and if , then 's have Beta distributions,See Devroye (1986) section XI.4, or Frigyik, Kapila &#38; Gupta (2010) .This distribution might also be called multivariate Beta distribution. In R, this function can be used as follows&#62; library(MCMCpack)&#62; alpha=c(2,2,5)&#62; x=seq(0,1,by=.05)&#62; vx=rep(x,length(x))&#62; vy=rep(x,each=length(x))&#62; vz=1-x-vy&#62; V=cbind(vx,vy,vz)&#62; D=ddirichlet(V, alpha)&#62; 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 &#38; Richards (1986)).