1978 TYS

A fair die is thrown once and random variable $X$ denotes the score obtained. Write down $\text{E}\left( X \right)$ and show that $\text{Var}\left( X \right)=\frac{35}{12}$. The die is thrown twice, and ${{X}_{1}}$and${{X}_{2}}$ obtained from each attempt.

Tabulate the probability distribution of $Y=\left| {{X}_{1}}-{{X}_{2}} \right|$, and find $E\left( Y \right)$.

The random variable $Z$ is defined by $Z={{X}_{1}}-{{X}_{2}}$. State with reasons whether or not

1. $E\left( {{Z}^{2}} \right)=E\left( {{Y}^{2}} \right)$.
2. $\text{Var}\left( Z \right)=\text{Var}\left( Y \right)$.