7.2.25 (MV) Suppose that (01,...,0k-1) ~ Dirichlet(a1, A2, ..., ak). (a) Prove that (@1,...,0k-2) Dirichlet(ai, a2, ...,ak-1 + ak). (Hint: In the inte- gral to integrate out 0k-1, make the transformation Ok-1 0k-1/(1-01-...-Ok-2).) (b) Prove that 01 Beta(ai, a2 + ... + ak). (Hint: Use part (a).) (c) Suppose (i1, ..., ik) is a permutation of (1, ...,k). Prove that (@i, ...,Oix-) ~ Dirichlet(Qij, Oi2, ..., Qir). (Hint: What is the Jacobian of this transformation?) (d) Prove that O; ~ Beta(di,a-i). (Hint: Use parts (b) and (c).) 7.2.25 (MV) Suppose that (01,...,0k-1) ~ Dirichlet(a1, A2, ..., ak). (a) Prove that (@1,...,0k-2) Dirichlet(ai, a2, ...,ak-1 + ak). (Hint: In the inte- gral to integrate out 0k-1, make the transformation Ok-1 0k-1/(1-01-...-Ok-2).) (b) Prove that 01 Beta(ai, a2 + ... + ak). (Hint: Use part (a).) (c) Suppose (i1, ..., ik) is a permutation of (1, ...,k). Prove that (@i, ...,Oix-) ~ Dirichlet(Qij, Oi2, ..., Qir). (Hint: What is the Jacobian of this transformation?) (d) Prove that O; ~ Beta(di,a-i). (Hint: Use parts (b) and (c).)