| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Session | Specimen |
| Marks | 4 |
| Topic | Cumulative distribution functions |
| Type | Distribution of order statistics |
| Difficulty | Standard +0.8 This question requires understanding of order statistics and the relationship between CDFs of independent random variables, which is conceptually sophisticated for A-level. Part (i) demands insight that P(T<t) = P(both X₁<t and X₂<t) = [F(t)]², requiring students to recognize and apply this non-standard technique. Parts (ii-iii) are routine once the CDF is established, but the initial conceptual leap and algebraic manipulation elevate this above typical Further Maths questions. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration |
**Question 9**
**(i)** $$\mathrm{P}(X < t) = \int_0^t 0.2\mathrm{e}^{-0.2x}\,\mathrm{d}x = \left[-\mathrm{e}^{-0.2x}\right]_0^t = 1 - \mathrm{e}^{-0.2t}$$ M1A1
$\mathrm{P}(\text{longer} < t) = \mathrm{P}(\text{shorter} < t) = 1 - \mathrm{e}^{-0.2t}$
$\Rightarrow \mathrm{P}(\text{both} < t) = (1-\mathrm{e}^{-0.2t})^2 = 1 - 2\mathrm{e}^{-0.2t} + \mathrm{e}^{-0.4t}$ M1
$$G(t) = \begin{cases} 1 - 2\mathrm{e}^{-0.2t} + \mathrm{e}^{-0.4t} & t > 0 \\ 0 & \text{otherwise} \end{cases}$$ (AG) A1 **[4 marks]**
**(ii)** $\mathrm{P}(\text{both served} \leq 10) = 1 - 2\mathrm{e}^{-2} + \mathrm{e}^{-4} = 0.748$ B1 **[1 mark]**
**(iii)** $(1-\mathrm{e}^{-0.2t})^2 = 0.25$ and $(1-\mathrm{e}^{-0.2t})^2 = 0.75$ M1A1
$\Rightarrow Q_1 = 3.4657...$ and $Q_3 = 10.0505...$ M1A1A1
Interquartile range $= 10.0505... - 3.4657... = 6.58$ (3sf) A1 **[6 marks]**9 The service time, $X$ minutes, for each customer in a post office is modelled by the probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x > 0 , \\ 0 & \text { otherwise } . \end{cases}$$
Two customers begin to be served independently at the same instant. The larger of the two service times is $T$ minutes.\\
(i) By considering the probability that both customers have been served in less than $t$ minutes, show that the cumulative distribution function of $T$ is given by
$$\mathrm { G } ( t ) = \begin{cases} 1 - 2 \mathrm { e } ^ { - 0.2 t } + \mathrm { e } ^ { - 0.4 t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
(ii) Find the probability that both customers are served within 10 minutes.\\
(iii) Find the value of the interquartile range of $T$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 Q9 [4]}}