1. (50 points) Ch. 13 Exercises : (a) #3.1

(b) #3.2

(c) #3.4

(d) #3.7

(e) #3.10 (a)

2. (30 points) Suppose that X and Y have joint p.m.f.

(a) Generate n = 10⁴ random variables Y₁, . . . , Y_n using a rejection sampling. You can use a Bernoulli distribution with parameter 1/2 as a candidate (proposal) distribution. Provide the (empirical) marginal p.m.f. of Y based on your random numbers and compare it with the (exact) marginal p.m.f. of Y .

(b) Generate n  =  10⁴  pairs of random variables {(X_i, Y_i)}n using a method of

composition with P (X = x Y = y) and P (Y = y). Provide your estimates

of P (X = x), x = 0, 1, 2 (the empirical marginal p.m.f. of X based on your random numbers) and compare them with the exact values of P (X = x) (the exact marginal p.m.f. of X).

(c) Generate n = 10⁴ pairs of random variables {(X_i, Y_i)}n using a Gibbs sampler.

Provide your estimates of P (X = x, Y = y), P (X = x), and P (Y = y), x = 0, 1, 2

and y = 0, 1 and compare them with the exact values.

3. (20  points)  Let  X₁, ..., X_n|θ  ∼  Bernoulli(θ),  and  assume  that  you  have  obtained  a

sample with x = Σn x_i = 710 successes in n = 1000 trials. Suppose that the value

of θ is unknown, and two statisticians A and B assign to θ the following different prior p.d.f.’s p_A(θ) and p_B(θ), respectively:

p_A(θ) = 2θ, for 0 < θ < 1, p_B(θ) = 3θ², for 0 < θ < 1.

(a) Find the posterior distribution that each statistician assigns to θ.

(b) Compute the posterior probability that θ is less than 0.4, P (θ < 0.4|x₁, . . . , x_n)

based on the prior p.d.f.  p_A(θ) using ○1   a grid approximation and ○2   a Monte

Carlo method, and compare your answer with the exact value using pbeta().

(c) Compute the posterior distribution of the log-odds ϕ = log ^θ based on the prior

p.d.f. p_B

(θ) using a Monte Carlo method.

1−θ

(d) Find the Bayes estimate for each statistician based on the following error loss function

L(θ, a) =

(θ − a)², for θ ≤ a,

2(θ − a)², for θ > a