Question: Exercise 5.6.9. Let G ~ Dirichlet(@1,...,ax) (so that G = (G1... ., GK-1)). Set a+ = an+. tax. (a) Show that E[Gx] = at/a+ and

Exercise 5.6.9. Let G ~ Dirichlet(@1,...,ax) (so that G = (G1... ., GK-1)). Set a+ = an+. tax. (a) Show that E[Gx] = at/a+ and Var[Gi] = a(at - a ) /(at(a+ + 1) ). [Hint: Use the beta representation in (5.28), and get the mean and variance from Table 1.1 on page 7.] (b) Find EGG,] for k # 1. (c) Show that Cop[Go, G] = -a / (at (a+ + 1)) for k * 1. (d) Show that Cov [G] = at(4+ +1) D(x) 1 (5.101) where a = (a1.....ag_1) and D(@) is the (K -1) x (K-1) diagonal matrix with the 4/'s on the diagonal: 0 0 DO) = (5.102) (K-1

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