The random variable \(X\) has a uniform distribution over the values \(\{ 1,2 , \ldots , n \}\). Show that \(\operatorname { Var } ( X )\) is given by \(\frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\).
The random variable \(Y\) has a uniform distribution over the values \(\{ 1,3,5 , \ldots , 2 n - 1 \}\). Using the result in part (a) or otherwise, show that \(\operatorname { Var } ( Y )\) is given by \(\frac { 1 } { 3 } \left( n ^ { 2 } - 1 \right)\).
Given that \(n = 100\), find the least value of \(k\) for which \(\mathrm { P } ( \mu - k \sigma \leqslant Y \leqslant \mu + k \sigma ) = 1\), where the mean and standard deviation of \(Y\) are represented by \(\mu\) and \(\sigma\) respectively.