| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2009 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moment generating functions |
| Type | Derive MGF from PDF |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question requiring integration of a piecewise exponential PDF to derive an MGF, then using it to find variance. It involves splitting the integral at x=0, handling convergence conditions carefully, and applying MGF differentiation techniques. More demanding than standard A-level but routine for S4 level. |
| Spec | 5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt use of identity for \(\sec^2 \alpha\) | M1 | using \(\pm \tan^2 \alpha \pm 1\) |
| Obtain \(1 + (m + 2)^2 - (1 + m^2)\) | A1 | absent brackets implied by subsequent correct working |
| Obtain \(4m + 4 = 16\) and hence \(m = 3\) | A1 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt subn in identity for \(\tan(\alpha + \beta)\) | M1 | using \(\pm \tan \alpha \pm \tan \beta / (1 \pm \tan \alpha \tan \beta)\) |
| Obtain \(\frac{5+3}{1-15}\) or \(\frac{m+2+m}{1-m(m+2)}\) | A1√ | following their \(m\) |
| Obtain \(-\frac{4}{9}\) | A1 3 | or exact equiv; condone absence of \(y = 0\) |
### (i)
Attempt use of identity for $\sec^2 \alpha$ | M1 | using $\pm \tan^2 \alpha \pm 1$ |
Obtain $1 + (m + 2)^2 - (1 + m^2)$ | A1 | absent brackets implied by subsequent correct working |
Obtain $4m + 4 = 16$ and hence $m = 3$ | A1 3 |
### (ii)
Attempt subn in identity for $\tan(\alpha + \beta)$ | M1 | using $\pm \tan \alpha \pm \tan \beta / (1 \pm \tan \alpha \tan \beta)$ |
Obtain $\frac{5+3}{1-15}$ or $\frac{m+2+m}{1-m(m+2)}$ | A1√ | following their $m$ |
Obtain $-\frac{4}{9}$ | A1 3 | or exact equiv; condone absence of $y = 0$ |3 The continuous random variable $X$ has probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} \mathrm { e } ^ { 2 x } & x < 0 \\ \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$
(i) Show that the moment generating function of $X$ is $\frac { 4 } { 4 - t ^ { 2 } }$, where $| t | < 2$, and explain why the condition $| t | < 2$ is necessary.\\
(ii) Find $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{OCR S4 2009 Q3 [9]}}