Suppose that (θ1,...,θK) has the (singular) DirichletK(α1,...,αK) distribution, withα1,...,αK>0. Focusing on the caseK= 3, prove the following results.(a) The marginal distribution ofθ1is Beta(α1,α2+α3).(b) The distribution ofθ1+θ2is Beta(α1+α2,α3).(c) Conditional distribution ofθ2/(1−θ1) givenθ1is Beta(α2,α3).(d) The covariance ofθ1andθ2isCov(θ1,θ2) =−α1α2α20(α0+ 1)whereα0=3∑j=1α
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