## Description

1. 10% of the emails I receive are spam. The automatic spam filter in my email system

correctly classifies an email with probability 0.99. What percentage of my emails in

the spam folder are expected to be genuine (non-spam)?

2. Suppose θ ∼ log − normal(0, 1), that is, log θ ∼ N(0, 1). Find the 95% equal-tail and

HPD credible intervals of θ. Compute the width of the two intervals. Which one do you

think is a better interval? (Hint: for HPD calculation, use the R package HDInterval).

3. Let yi

|θ ∼iid Exp (θ), for i = 1, . . . , n. (p(y|θ) = θe−θy). Assume the prior distribution

for θ to be Gamma(a, b), that is, p(θ) = b

a/Γ(a)θ

a−1

exp{−bθ}. Find the posterior

distribution of θ.

4. Find the posterior predictive distribution of a future observation in problem 3.

5. Let yi

|θ ∼iid Uniform (0, θ), for i = 1, . . . , n. Assume the prior distribution for θ to

be P areto(a, b), where p(θ) = bab/θb+1 for θ > a and 0 otherwise. Find the posterior

distribution of θ.