  This paper describes how one can use the well-known Bayesian prior to
posterior analysis of the Dirichlet process, and less known results for the
gamma process, to address the formidable problem of assessing the distribution
of linear functionals of Dirichlet processes. In particular, in conjunction
with a gamma identity, we show easily that a generalized Cauchy-Stieltjes
transform of a linear functional of a Dirichlet process is equivalent to the
Laplace functional of a class of, what we define as, beta-gamma processes. This
represents a generalization of the Markov-Krein identity for mean functionals
of Dirichlet processes. A prior to posterior analysis of beta-gamma processes
is given that not only leads to an easy derivation of the Markov-Krein
identity, but additionally yields new distributional identities for gamma and
beta-gamma processes. These results give new explanations and intepretations of
exisiting results in the literature. This is punctuated by establishing a
simple distributional relationship between beta-gamma and Dirichlet processes.
