N1978 P2 Q12

A fair die is thrown once and random variable \(X\) denotes the score obtained. Write down \(\mathrm{E}\left( X \right)\) and show that \(\mathrm{Var}\left( X \right)=\frac{35}{12}\). 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. \(\mathrm{Var}\left( Z \right)=\mathrm{Var}\left( Y \right)\).