| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Topic | Moment generating functions |
| Type | MGF series expansion |
| Difficulty | Standard +0.8 This question requires computing an MGF integral that yields a non-standard function (sinh t/t), then manipulating the series expansion of sinh t to extract moments. While the integration is straightforward, recognizing how to extract E(X^4) from the series expansion requires careful coefficient matching and understanding of MGF theory beyond routine application. This is moderately challenging for Further Maths students. |
| Spec | 5.03c Calculate mean/variance: by integration |
Total: 9 marks
**Part (i):**
Attempt $\int f(x)e^{tx}dx$, correct limits (could be later) **M1**
$\int_{-1}^{1} f(x)e^{tx}dx = \int_{-1}^{1} \frac{1}{2}e^{tx}dx$ **A1** Correct expression
$= \dfrac{1}{2t}\left[e^{tx}\right]_{-1}^{1}$ **A1** Correct integral **AG**
$= \dfrac{e^t - e^{-t}}{2t} = \dfrac{\sinh t}{t}$ **A1** Correctly obtain given answer. SC: Using formula for MGF of uniform distribution from formula book: use of formula and substituting $a=-1$, $b=1$ M1; correctly obtain given answer A1. Max 2/4.
**Part (ii):**
$\dfrac{1}{t}\!\left(t + \dfrac{t^3}{3!} + \dfrac{t^5}{5!} + \ldots\right)$ **M1** Correct expansion of $\sinh t$ used
$= 1 + \dfrac{t^2}{6} + \dfrac{t^4}{120} + \ldots$ **A1** Correct after division by $t$ (at least 3 terms) soi
$E(X^2) = M''_X(0) = 2! \times \text{coeff of } t^2 = \frac{1}{3}$ **M1** Use $2!\times$coeff of $t^2$ or attempt to diff twice
$\text{Var}(X) = \frac{1}{3}$ **A1** $\frac{1}{3}$, distinction between $E(X^2)$ and $\text{Var}(X)$ made.
$E(X^4) = 4! \times \text{coeff of } t^4 = \frac{1}{5}$ **A1** or $E(X)$ (or $M'_X(0)$) stated to be zero; correctly obtain $1/5$.
Total: **9 marks**6 The random variable $X$ has a uniform distribution on the interval $[ - 1,1 ]$, so that its probability density function is given by
$$f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
(i) Show from the definition of the moment generating function that the moment generating function of $X$ is $\frac { \sinh t } { t }$.\\
(ii) By using the series expansion of $\sinh t$, find the variance of $X$ and the value of $\mathrm { E } \left( X ^ { 4 } \right)$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q6 [11]}}