| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Topic | Probability Generating Functions |
| Type | Determine constant in PGF |
| Difficulty | Standard +0.3 This is a straightforward PGF question requiring standard techniques: finding k using G(1)=1, identifying the modal value from coefficients, using the independence property for sums, and applying G'(1) and G''(1) formulas for mean and variance. All steps are routine applications of PGF theory with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
(i)(a) $k(5 + 3 + 2) = 1 \Rightarrow k = \frac{1}{10}$ **B1** [1]
(i)(b) Modal value is $-1$. **B1** [1]
(ii)(a) $G_Y(t) = \frac{1}{100}(5t^{-1} + 3 + 2t^2)^2$ (ft on $k$ value and also in (b): 1st two lines.) **B1**✓ [1]
(ii)(b) $G_Y'(t) = \frac{1}{50}(5t^{-1} + 3 + 2t^2)(-5t^{-2} + 4t)$ **M1A1**✓
$E(Y) = G_Y''(1) = -\frac{10}{50} = -\frac{1}{5}$ **B1**
$G_Y''(t) = \frac{1}{50}\{(-5t^{-2} + 4t)^2 + (5t^{-1} + 3 + 2t^2)(10t^{-3} + 4)\}$ **M1A1**✓
$G_Y''(1) = \frac{1}{50}(1 + 10 \times 14) = \frac{141}{50} = \sigma^2 + \frac{1}{25} + \frac{1}{5}$ **A1B1**
(B1 for their $\sigma^2 + \mu^2 = \mu$.)
$\Rightarrow \sigma^2 = \frac{129}{50} = 2.58$ (AWRT 2.58) **A1** [8]5 The discrete random variable $X$ has probability generating function given by
$$\mathrm { G } _ { X } ( t ) = k \left( 5 t ^ { - 1 } + 3 + 2 t ^ { 2 } \right) ,$$
where $k$ is a constant.\\
(i) Find
\begin{enumerate}[label=(\alph*)]
\item the value of $k$,
\item the modal value of $X$.\\
(ii) The random variables $X _ { 1 }$ and $X _ { 2 }$ are independent observations of $X$.\\
(a) Write down the probability generating function of $Y$, where $Y = X _ { 1 } + X _ { 2 }$.\\
(b) Use your answer to part (ii)(a) to find $\mathrm { E } ( Y )$ and $\operatorname { Var } ( Y )$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q5 [8]}}