Find the mean and the variance of

(i)     \({X_1} + {X_2}\) ;

(ii)     \(3{X_1}\);

(iii)     \({X_1} + {X_2} - {X_3}\) ;

(iv)     \(\bar X = \frac{{({X_1} + {X_2} + ... + {X_n})}}{n}\).

Find \({\text{E}}(X_1^2)\) in terms of \(\mu \) and \(\sigma \) .

(i)     \(2\mu ,{\text{ }}2{\sigma ^2}\)     A1A1

(ii)     \(3\mu ,{\text{ }}9{\sigma ^2}\)     A1A1

(iii)     \(\mu ,{\text{ }}3{\sigma ^2}\)     A1A1

(iv)     \(\mu ,{\text{ }}\frac{{{\sigma ^2}}}{n}\)     A1A1

Note: If candidate clearly and correctly gives the standard deviations rather than the variances, give A1 for 2 or 3 standard deviations and A1A1 for 4 standard deviations.

[8 marks]

\({\text{Var}}({X_1}) = {\text{E}}(X_1^2) - {\left( {{\text{E}}({X_1})} \right)^2}\)     (M1)

\({\sigma ^2} = {\text{E}}(X_1^2) - {\mu ^2}\)     (A1)

\({\text{E}}(X_1^2) = {\sigma ^2} + {\mu ^2}\)     A1

[3 marks]

This was very well answered indeed with very many candidates gaining full marks including, pleasingly, part (b). Candidates who could not do question 2, struggled on the whole paper.

This was very well answered indeed with very many candidates gaining full marks including, pleasingly, part (b). Candidates who could not do question 2, struggled on the whole paper.