| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Marks | 9 |
| Topic | Moment generating functions |
| Type | MGF of transformed variable |
| Difficulty | Standard +0.8 This is a comprehensive MGF question requiring multiple techniques: deriving an MGF from a pdf via integration, using MGF derivatives for moments, finding MGF of transformed variables, and combining independent MGFs. While each individual step is methodical, the multi-part structure and requirement to work with MGFs of differences (less routine than sums) places this above average difficulty for Further Maths statistics. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration |
From Question 1 in the mark scheme:
**(i)** $M_X(t) = \int_0^\infty e^{tx} k e^{-kx}\,dx$ (Limits required) **M1**
$= k\int_0^\infty e^{(t-k)x}\,dx = k\int_0^\infty e^{-(k-t)x}\,dx$ (Limits not required) **M1**
$= \frac{-k}{k-t}\left[e^{-(k-t)x}\right]_0^\infty = \frac{k}{k-t}$ AG **A1**
**(ii)** $M_X'(t) = \frac{k}{(k-t)^2} \Rightarrow E(X) = M_X'(0) = \frac{1}{k}$ **M1 A1**
$M_X''(t) = \frac{2k}{(k-t)^3} \Rightarrow E(X^2) = M_X''(0) = \frac{2}{k^2}$ **M1 A1**
(A1 ft if double sign error when differentiating twice, but CAO) **A1**
**Alternatively:**
$M_X(t) = \left(1 - \frac{t}{k}\right)^{-1} = 1 + \frac{t}{k} + \frac{t^2}{k^2} + \ldots = 1 + \frac{1}{k}t + \frac{2}{k^2}t^2 + \ldots$ **M1 A1**
$E(X) = \frac{1}{k}$ **A1**
$E(X^2) = \frac{2}{k^2} \Rightarrow \text{Var}(X) = \frac{2}{k^2} - \left(\frac{1}{k}\right)^2 = \frac{1}{k^2}$ **M1 A1**5 The random variable $X$ has probability density function $\mathrm { f } ( x )$, where
$$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$
and $k$ is a positive constant.\\
(i) Show that the moment generating function of $X$ is $\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k$.\\
(ii) Use the moment generating function to find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.\\
(iii) Show that the moment generating function of $- X$ is $k ( k + t ) ^ { - 1 }$.\\
(iv) $X _ { 1 }$ and $X _ { 2 }$ are two independent observations of $X$. Use the moment generating function of $X _ { 1 } - X _ { 2 }$ to find the value of $\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q5 [9]}}