Consider the population described by the probability distribution shown below. The

random variable $x$ is observed twice with independent observation.

$($a$)$ List all possible samples and the corresponding values of sample mean $(\stackrel{}{x}),$ sample

range, and the probability.

$($b$)$ Construct the sampling distribution table of sample mean.

$($c$)$ Construct the sampling distribution table of sample range.

$($d$)$ What is the probability that $\stackrel{}{x}$ is $3$ or larger?

$($e$)$ What is the probability that range$($x$)$ is $2$ or larger?