| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Linear transformation to achieve target parameters |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of normal distributions, setting up and solving simultaneous equations from mean and variance conditions, then applying the distribution of X-Y to find σ². While the concepts are standard S3 material, the multi-step algebraic manipulation and integer constraint make it moderately challenging, above average difficulty but not requiring exceptional insight. |
| Spec | 5.04b Linear combinations: of normal distributions |
5 The independent random variables $X$ and $Y$ have distributions $\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)$ and $\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)$ respectively. The random variable $a X + b Y$, where $a$ and $b$ are constants, has the distribution $\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)$.\\
(i) Given that $a$ and $b$ are integers, find the value of $a$ and the value of $b$.\\
(ii) Given that $\mathrm { P } ( X > Y ) = 0.966$, find $\sigma ^ { 2 }$.
\hfill \mbox{\textit{OCR S3 2016 Q5 [11]}}