Consider the Population Described by the Probability Distribution Below $\begin{array}{c|c|c|c}x & 2 & 5 & 7 \\\hline p(x) & .2 & .5 & .3\end{array}$

Essay

Consider the population described by the probability distribution below.

$\begin{array}{c|c|c|c}x & 2 & 5 & 7 \\\hline p(x) & .2 & .5 & .3\end{array}$

The random variable $x$ is observed twice. The observations are independent. The different samples of size 2 and their probabilities are shown below.

$\begin{array}{c}\begin{array}{cc}\hline \text { Sample } &\text { Probability } \\\hline 2,2 & .04 \\2,5& .10 \\2,7 & .06\\\hline \end{array}\begin{array}{cc}\hline \text { Sample } & \text { Probability } \\\hline 5,2 & .10 \\5,5 & .25 \\5,7 & .15 \\\hline \end{array}\begin{array}{cc}\hline \text { Sample } & \text { Probability } \\\hline 7,2 & .06 \\7,5 & .15 \\7,7 & .09 \\\hline \end{array}\end{array}$



Find the sampling distribution of the sample mean $\bar { x }$ .