## Description

1. (90pts total, equally weighted) We write X ∼ Be(α, β) if X has the beta distribution with parameters

α > 0 and β > 0, that is, its pdf is

p(x | α, β) = Be(x | α, β) = 1

B(α, β)

x

α−1

(1 − x)

β−1

,

where B(α, β) is the beta function. Suppose that we want to generate samples from the following target

density (known up to a constant)

f(x; α, β) ∝ x

α−1

(1 − x)

β−1

,

with α = 2.7, β = 6.3. Work on the following questions.

(a) Plot the densities of f(x; α = 2.7, β = 6.3) and the Uniform distribution U(0, 1). According to the

rejection sampling approach, sample from the beta distribution using the U(0, 1) pdf as an enveloping

function.

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(b) Plot the histogram of the points that fall in the acceptance region. Do this for a simulation size of 102

and 105 and report your acceptance ratio. Compare the ratios and histograms.

(c) Instead of using the uniform distribution, using Be(2, 6) as the enveloping function. Then repeat the

tasks (a) and (b)

(d) Provide the four histograms from Tasks (2) and (3) using the U(0, 1) and Be(2, 6) enveloping proposals.

Provide the acceptance ratios. Provide commentary.

(e) Use a uniform distribution as a symmetric proposal, and implement the Metropolis algorithm. You

need to tune the width of the interval around the current value to achieve good performance. Plot the

traceplot and histogram.

(f) Use a normal distribution as a symmetric proposal, and implement the Metropolis algorithm. You

need to tune the width of the interval around the current value to achieve good performance. Plot the

traceplot and histogram.

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