| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Topic | Exponential Distribution |
| Type | Link Poisson to exponential |
| Difficulty | Standard +0.3 This is a standard textbook exercise linking Poisson and exponential distributions. Part (i)(a) requires recognizing that P(T<t) = P(at least one goal in time t) = 1-P(0 goals), a direct application of the memoryless property. Part (i)(b) is routine differentiation. Part (ii) involves solving exponential equations for quartiles—straightforward algebra with logarithms. While it requires understanding the connection between distributions, the execution is mechanical with no novel insight needed. |
| Spec | 5.02i Poisson distribution: random events model5.03a Continuous random variables: pdf and cdf5.03f Relate pdf-cdf: medians and percentiles |
**(i)(a)** $P(0 \text{ goals in } t \text{ minutes}) = e^{-\frac{1}{24}t}$
$\left(P(T > t) = P(0 \text{ goals in } [0,t]) = e^{-\frac{1}{24}t}\right) \Rightarrow P(T < t) = F(t) = 1 - e^{-\frac{1}{24}t},\ (t \geq 0).$ (AG) — B1, B1 [2]
**(b)** $f(t) = F'(t) = \frac{1}{24}e^{-\frac{1}{24}t},\ (t \geq 0).$ — M1A1 [2]
**(ii)** $1 - e^{-\frac{1}{24}q_1} = \frac{1}{4} \Rightarrow e^{-\frac{1}{24}q_1} = \frac{3}{4} \Rightarrow q_1 = 24\ln\left(\frac{4}{3}\right)$ — M1A1
Similarly: $q_3 = 24\ln 4$ — A1
$\text{IQR} = 24\left(\ln 4 - \ln\frac{4}{3}\right) = 24\ln 3$ (Accept 26.4) — A1$\checkmark$ [4]
**[Total: 8]**7 The number of goals scored by a hockey team in an interval of time of length $t$ minutes follows a Poisson distribution with mean $\frac { 1 } { 24 } t$. The random variable $T$ is defined as the length of time, in minutes, between successive goals.
\begin{enumerate}[label=(\roman*)]
\item (a) Show that $\mathrm { P } ( T < t ) = 1 - \mathrm { e } ^ { - \frac { 1 } { 24 } t }$ for $t \geqslant 0$.\\
(b) Hence find the probability density function of $T$.
\item Find the exact value of the interquartile range of $T$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2010 Q7 [8]}}