3. (29 pt) $X_1, \dots, X_4$ are independent Unif(0,$\theta$) random variables, where $\theta \in [0, 10]$. We would
like to perform Bayesian inferences with prior distribution with prior density $\pi(\theta) = c \times \theta \times$
$1(0 \le \theta \le 10)$ for some constant c. Assume that observations are s = {2.8, 0.3, 4.0, 3.6}.
a. (2pt) What is the value of c that ensures $\pi(\theta)$ is a proper probability distribution?
b. (8pt) Please derive the posterior distribution of $\theta$ based on the given data set, step by
step (the solution must not contain generic r.v.'s $X_1$-$X_4$, and directly rely on the given
data set)
b.1 (2pt) The likelihood function $\mathcal{L}(\theta; s) = ?$
b.2 (2pt) $\pi(\theta) \times \mathcal{L}(\theta; s) = ?$
b.3 (2pt) $\int \pi(\theta) \times \mathcal{L}(\theta; s)d\theta = ?$
b.4 (2pt) $\pi(\theta|s) = ?$
c. (2pt) What is the MLE estimation, based on the likelihood function.
d. (8pt) Based on the posterior distribution, please derive the posterior mode, posterior
mean and 80% posterior HPD credible interval.
e. (7pt) Consider $H_0 = \theta \le 5$ vs $H_a: \theta > 5$.
e.1 (2pt) What is the prior probability for null hypothesis?
e.2 (2pt) What is the posterior probability for null hypothesis?
e.3 (5pt) What is the Bayesian factor in favor of null hypothesis, and what is your
testing conclusion based on Bayes Factor?
