The Probability Distribution Shown Below Describes a Population of Measurements $\begin{array}{c|ccc}\mathrm{x} & 0 & 2 & 4 \\\hline \mathrm{p}(\mathrm{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}$

The probability distribution shown below describes a population of measurements. $\begin{array}{c|ccc}\mathrm{x} & 0 & 2 & 4 \\\hline \mathrm{p}(\mathrm{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}$

Suppose that we took repeated random samples of $n = 2$ observations from the population described above. Which of the following would represent the sampling distribution of the sample mean?

A) $\begin{array}{c|ccc}\bar{x} & 0 & 2 & 4 \\\hline p(\bar{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}$

B) $\begin{array}{c|ccccc}\bar{x} & 0 & 1 & 2 & 3 & 4 \\\hline \mathrm{p}(\bar{x}) & 2 / 9 & 2 / 9 & 1 / 9 & 2 / 9 & 2 / 9\end{array}$

C) $\begin{array}{c|ccccc}\overline{\mathrm{x}} & 0 & 1 & 2 & 3 & 4 \\\hline \mathrm{p}(\overline{\mathrm{x}}) & 1 / 9 & 2 / 9 & 3 / 9 & 2 / 9 & 1 / 9\end{array}$

D) $\begin{array}{c|ccccc}\overline{\mathrm{x}} & 0 & 1 & 2 & 3 & 4 \\\hline \mathrm{p}(\overline{\mathrm{x}) } & 1 / 5 & 1 / 5 & 1 / 5 & 1 / 5 & 1 / 5\end{array}$