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Rules: You must unite your name and the East .5 digits of your student ID on the submitted work. Suppose that heights of individuals in a population...

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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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Rules: You must unite your name and the East .5 digits of your student ID on the
submitted work. If you consult anybody other than the instructor or the teaching assistants
regarding the assignment, or used any resources other than the reference books listed in the
syllabus, you must report that in your submission. 1. Suppose that heights of individuals in a population follow the Normal{p, oz) distribu—
tion. Suppose also that a random sample of n = 25 individuals from this popuiation
yielded sample mean 1—? = 155 centimeters and sample standard deviation s 2 ll]
centimeters. Assume the following prior specification for (3;, 0'2). me? ~ N{150,02/10)
0'2 N Inv—xz(4,15). [Using the same notations as in lecture Note 6.] (a) Find the conditional posterior distribution of ,u. given (0'2, 1'1, . . .1 1‘"). (b) Find the marginal posterior distribution of p.
(c) Find the 95% highest posterior density (HPD} credible region for ,u.
(d) Find a 95% posterior credible interval for 0'2. (e) [f a new individual is randomly selected from the population, what is the posterior
probability that this person’s height will be between 15!] and 160 centimeters '? 2. Suppose that (91,. . . ,BK) has the(sing1.11ar] Dirichletgmrl, . .. , UK) distribution, with
{21, .. . , tag :2: D. Focusing on the case K 2 F3, prove the following results.
(a) The marginal distribution of 9'1 is Betalzicri, [1'2 + £23).
(b) The distribution of 31 + 62 is Beta(cr1 + :12, 0:3).
(c) Conditional distribution of 32f“. — 31) given 91 is Beta(org, 043). (d) The covariance of 61 and 32 is emote) = :11”: {an + 1} 3
where an: E 03. 3. Suppose that data (XI, .. . ,XK) follows a. (singular) Multinomial distribution with

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