Date | November 2012 | Marks available | 8 | Reference code | 12N.3sp.hl.TZ0.2 |
Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
Command term | Find | Question number | 2 | Adapted from | N/A |
Question
The n independent random variables X1,X2,…,Xn all have the distribution N(μ,σ2).
Find the mean and the variance of
(i) X1+X2 ;
(ii) 3X1;
(iii) X1+X2−X3 ;
(iv) ˉX=(X1+X2+...+Xn)n.
Find E(X21) in terms of μ and σ .
Markscheme
(i) 2μ, 2σ2 A1A1
(ii) 3μ, 9σ2 A1A1
(iii) μ, 3σ2 A1A1
(iv) μ, σ2n 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]
Var(X1)=E(X21)−(E(X1))2 (M1)
σ2=E(X21)−μ2 (A1)
E(X21)=σ2+μ2 A1
[3 marks]
Examiners report
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.