1. Let $X_1, ..., X_n$ be a random sample from a continuous distribution with the cumulative
distribution function (CDF)
$$F_X(x; \theta) = \begin{cases}
1 - \left(\frac{2}{x}\right)^\theta, & x \ge 3, \\
0, & \text{otherwise}
\end{cases}$$
Here, $\theta > 0$ is an unknown parameter.
(a) Find the maximum likelihood estimator of $\theta$, $\hat{\theta}_{ML}$. Show that $\hat{\theta}_{ML}$ maximizes the
likelihood function.
(b) Find the Cramer-Rao lower bound for the variance of an unbiased estimator of $\theta$.
(c) Now assume the sample size is $n = 100$ and the observed data are such that the
maximum likelihood estimate $\hat{\theta} = 2.5$. Using these data and asymptotic nor-
mality of the maximum likelihood estimator $\hat{\theta}_{ML}$, construct an approximate 95%
confidence interval for $\theta$.
(d) Consider the exponential prior distribution for the parameter $\theta$ with the proba-
bility density function (PDF):
$$f_\theta(\theta) = \begin{cases}
e^{-\theta}, & \theta > 0, \\
0, & \text{otherwise}.
\end{cases}$$
Find the posterior distribution of $\theta$ using the same observed data as in (c). Find
the posterior mean of $\theta$. Is it close to the maximum likelihood estimate $\hat{\theta} = 2.5$?
Explain why or why not. Hint: $a^x = e^{x \ln a}$ for $a > 0$.
(e) We would like to test the null hypothesis $H_0: \theta = 2$ against the alternative
$H_a: \theta > 2$. Construct the test using asymptotic normality of $\hat{\theta}_{MLE}$. Compute
the p-value of this test using the same observed data as in (c). Do we reject $H_0$
at the 1% significance level?
