N1984 P2 Q11

The discrete random variable \(X\) takes values \(-1\), \(0\), \(1\) with probabilities \(\frac{1}{4}\), \(\frac{1}{2}\), \(\frac{1}{4}\) respectively. The variable \(\bar{X}\) is the mean of a random sample of \(3\) values of \(X\) (i.e. \({{X}_{1}}\), \({{X}_{2}}\) and \({{X}_{3}}\) are independent random variables).

Tabulate the probability distribution of \(\bar{X}\), and use your values to calculate \(\mathrm{Var}\left( {\bar{X}} \right)\).

Hence verify that \(\mathrm{Var}\left( {\bar{X}} \right)=\frac{1}{3}\mathrm{Var}\left( X \right)\) in this case.