| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Moderate -0.8 This is a straightforward application of standard uniform distribution formulas. Students need to recall E(T) = (α+β)/2 and Var(T) = (β-α)²/12, solve two simultaneous equations, then apply the uniform PDF for a probability calculation. All steps are routine with no problem-solving insight required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(T) = \frac{\alpha + \beta}{2} = 2 \Rightarrow \alpha + \beta = 4\) | B1 | |
| \(\text{Var}(T) = \frac{(\beta - \alpha)^2}{12} = 3 \Rightarrow (\beta - \alpha)^2 = 64\) | B1 | |
| \(\alpha = -2, \beta = 6\) | M1 A1 A1 | (5) |
| \(P(T < 3.4) = \frac{1}{8} \times (5.4)\) | M1 | |
| \(= 0.675\) | A1 | (2) |
| [7] |
| Answer | Marks |
|---|---|
| Item | Guidance |
| (a) | 1st B1: \(\alpha + \beta = 4\) o.e. |
| 2nd B1: \((\beta - \alpha)^2 = 64\) o.e. Allow \((\beta - \alpha) = \pm 8\) or \((\beta - \alpha) = -8\) or \(3(\beta - \alpha)^2 = 192\) | |
| May be implied by a correct equation in one variable | |
| M1: Correct processes to obtain a correct equation in one variable. Allow one slip. | |
| e.g. \((\beta - [4-\beta])^2 = 64\) or \(2\beta = 12\) or \(4\alpha^2 - 16\alpha - 48 = 0\) or \((2-\alpha)^2 = 16\) | |
| 1st A1: \(\alpha = -2,\) | |
| 2nd A1: \(\beta = 6\) | |
| If both correct answers only appear then this implies all 5 marks. | |
| (b) | M1: \(\frac{1}{\pm \text{their } "(\beta - \alpha)"} \times (3.4 - \text{'their } \alpha')\) If their expression is \(-\nu e\) or \(> 1\) then M0 |
| A1: 0.675 or exact equivalent e.g. \(\frac{27}{40}\) |
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(T) = \frac{\alpha + \beta}{2} = 2 \Rightarrow \alpha + \beta = 4$ | B1 | |
| $\text{Var}(T) = \frac{(\beta - \alpha)^2}{12} = 3 \Rightarrow (\beta - \alpha)^2 = 64$ | B1 | |
| $\alpha = -2, \beta = 6$ | M1 A1 A1 | (5) |
| $P(T < 3.4) = \frac{1}{8} \times (5.4)$ | M1 | |
| $= 0.675$ | A1 | (2) |
| | [7] | |
**Notes:**
| Item | Guidance |
|---|---|
| (a) | 1st B1: $\alpha + \beta = 4$ o.e. |
| | 2nd B1: $(\beta - \alpha)^2 = 64$ o.e. Allow $(\beta - \alpha) = \pm 8$ or $(\beta - \alpha) = -8$ or $3(\beta - \alpha)^2 = 192$ |
| | | May be implied by a correct equation in one variable |
| | M1: Correct processes to obtain a correct equation in one variable. Allow one slip. |
| | | e.g. $(\beta - [4-\beta])^2 = 64$ or $2\beta = 12$ or $4\alpha^2 - 16\alpha - 48 = 0$ or $(2-\alpha)^2 = 16$ |
| | 1st A1: $\alpha = -2,$ |
| | 2nd A1: $\beta = 6$ |
| | | If both correct answers only appear then this implies all 5 marks. |
| (b) | M1: $\frac{1}{\pm \text{their } "(\beta - \alpha)"} \times (3.4 - \text{'their } \alpha')$ If their expression is $-\nu e$ or $> 1$ then M0 |
| | A1: 0.675 or exact equivalent e.g. $\frac{27}{40}$ |
---\begin{enumerate}
\item The continuous random variable $T$ is uniformly distributed on the interval $[ \alpha , \beta ]$ where $\beta > \alpha$
\end{enumerate}
Given that $\mathrm { E } ( T ) = 2$ and $\operatorname { Var } ( T ) = \frac { 16 } { 3 }$, find\\
(a) the value of $\alpha$ and the value of $\beta$,\\
(b) $\mathrm { P } ( T < 3.4 )$\\
\hfill \mbox{\textit{Edexcel S2 2014 Q3 [7]}}