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

question 43

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

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

$\begin{array}{cc}\hline \text { Sample } & \text { Probability } \\\hline 7,2 & .06 \\7,5 & .15 \\7,7 & .09 \\\hline\end{array}$
Find the sampling distribution of the sample mean $\bar { x }$ .

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Definitions:

Substitution Reaction

A chemical reaction in which an atom or group of atoms in a molecule is replaced by another atom or group of atoms.

Mechanism

The detailed process by which a chemical reaction occurs, at the level of molecular structure.

Protic Solvents

Solvents that have hydrogen atoms capable of forming hydrogen bonds, often used in reactions requiring proton transfer.

Alkenes

Hydrocarbons containing at least one carbon-carbon double bond, characterized by their unsaturated nature.