| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Topic | Moment generating functions |
| Type | Verify MGF convergence condition |
| Difficulty | Standard +0.3 This is a standard MGF question requiring integration by parts and routine manipulation. Part (i) is a guided calculation with a helpful hint, part (ii) asks for the convergence condition (t<2), and part (iii) uses standard MGF properties. While it involves multiple steps, all techniques are textbook applications with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
(i) $\int_0^\infty 4xe^{-2x}e^{tx}\,\mathrm{d}x = \int_0^\infty 4xe^{-(2-t)x}\,\mathrm{d}x$
$= \left[\dfrac{4xe^{-(2-t)x}}{(t-2)}\right]_0^\infty + \int_0^\infty \dfrac{4e^{-(2-t)x}}{2-t}\,\mathrm{d}x$
$= \left[\dfrac{-4e^{-(2-t)x}}{(2-t)^2}\right]_0^\infty = \dfrac{4}{(2-t)^2}$
- M1: Attempt $\int e^{tx}f(x)\,\mathrm{d}x$, limits somewhere
- A1: Combine into single $e$ term
- M1: Use parts, right way round
- A1: Correct indefinite integral
- A1: Correct final answer, cwo, allow $(t-2)^2$ but must use integral that visibly converges, or otherwise indicate the issue
**[5]**
(ii) $t < 2$
- B1
**[1]**
(iii) $\left[\dfrac{4}{(2-t)^2}\right]^3 = \dfrac{64}{(2-t)^6}$
$= \left(1-\tfrac{1}{2}t\right)^{-6} = 1 + 3t + \tfrac{21}{4}t^2 + \ldots$
$\text{E}(Y) = \mathbf{3}$
$\text{E}(Y^2)/2 = 21/4$ so $\text{E}(Y^2) = 10.5$
$\text{Var}(Y) = 10.5 - 3^2 = \mathbf{1.5}$
- M1: $[M_X(t)]^3$; [Not cubed: M0A0 M1A0 M1A0]
- A1
- M1: Series expansion *or* differentiate once; $M'(t) = \dfrac{384}{(2-t)^7}$, $M''(t) = \dfrac{2688}{(2-t)^8}$
- A1: $\text{E}(Y) = 3$ correctly obtained or implied
- M1: $2 \times \text{coeff of }t^2$ *or* $M''(0) - [M'(0)]^2$
- A1: $\text{Var}(Y) = 1.5$ or exact equivalent, cwo
**[6]**6 A continuous random variable $X$ has probability density function
$$\mathrm { f } ( x ) = \begin{cases} 4 x \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } . \end{cases}$$
(i) Show that the moment generating function $\mathrm { M } _ { X } ( t )$ of $X$ is $\frac { 4 } { ( 2 - t ) ^ { 2 } }$. You may assume that $x \mathrm { e } ^ { - k x } \rightarrow 0$ as $x \rightarrow + \infty$.\\
(ii) What condition on $t$ is needed in finding $\mathrm { M } _ { X } ( t )$ ?\\
(iii) $Y$ is the sum of three independent observations of $X$. Find the moment generating function of $Y$, and use your answer to find $\operatorname { Var } ( Y )$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q6 [6]}}