# ISYE 6420 A/MSA Homework 1 solution

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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.