| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Linear transformation to achieve target parameters |
| Difficulty | Standard +0.3 This is a straightforward application of linear combinations of normal distributions. Part (i) requires simple algebra with expectations, part (ii) uses standard formulas for variance of independent variables, and part (iii) asks for a brief conceptual comment. All techniques are routine for S3 level with no novel problem-solving required. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(E(T) = E(X) + \lambda E(Y) \Rightarrow 100 = 45 + 33\lambda \Rightarrow \lambda = \frac{5}{3}\) AG | M1, A1 | Use \(E(X + \lambda Y)\); 2 aef |
| (ii) \(\text{Var}(T) = \text{Var}(X) + (\lambda)^2\text{Var}(Y) = 256\); \(T \sim N(100, 256)\) | M1, A1, B1√ | 3 ft variance |
| (iii) Same student for \(X\) and \(Y\) so independence unlikely | B1 | 1 Sensible reason |
**(i)** $E(T) = E(X) + \lambda E(Y) \Rightarrow 100 = 45 + 33\lambda \Rightarrow \lambda = \frac{5}{3}$ AG | M1, A1 | Use $E(X + \lambda Y)$; 2 aef
**(ii)** $\text{Var}(T) = \text{Var}(X) + (\lambda)^2\text{Var}(Y) = 256$; $T \sim N(100, 256)$ | M1, A1, B1√ | 3 ft variance
**(iii)** Same student for $X$ and $Y$ so independence unlikely | B1 | 1 Sensible reason1 The marks obtained by a randomly chosen student in the two papers of an examination are denoted by the random variables $X$ and $Y$, where $X \sim \mathrm {~N} ( 45,81 )$ and $Y \sim \mathrm {~N} ( 33,63 )$. The student's overall mark for the examination, $T$, is given by $T = X + \lambda Y$, where the constant $\lambda$ is chosen such that $\mathrm { E } ( T ) = 100$.\\
(i) Show that $\lambda = \frac { 5 } { 3 }$.\\
(ii) Assuming that $X$ and $Y$ are independent, state the distribution of $T$, giving the values of its parameters.\\
(iii) Comment on the assumption of independence.
\hfill \mbox{\textit{OCR S3 2007 Q1 [6]}}