| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Topic | Sum of Poisson processes |
| Type | Minimum time or stock level |
| Difficulty | Challenging +1.2 This question combines a standard proof (sum of Poisson distributions) with routine applications of Poisson probability calculations. Part (i) is a well-known result requiring algebraic manipulation of probability generating functions or direct convolution. Parts (ii) and (iii) involve straightforward substitution into Poisson formulas and cumulative probability calculations. While it requires multiple techniques and careful arithmetic, it contains no novel insights and follows standard Further Maths Statistics patterns. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02n Sum of Poisson variables: is Poisson |
**(i) Either**
$P(X + Y = r) = \sum_{s=0}^{r} \frac{e^{-\lambda}\lambda^s}{s!} \cdot \frac{e^{-\mu}\mu^{r-s}}{(r-s)!}$ — M1
$= \frac{e^{-(\lambda+\mu)}}{r!}\sum_{s=0}^{r}\frac{r!}{s!(r-s)!}\lambda^s\mu^{r-s}$ — M1
$= \frac{e^{-(\lambda+\mu)}}{r!}(\lambda+\mu)^r \Rightarrow X + Y \sim \text{Po}(\lambda+\mu)$ — A1
**Or** PGF of $X$: $G_x(t) = e^{\lambda(t-1)}$; PGF of $Y$: $G_y(t) = e^{\mu(t-1)}$ — M1
PGF of $Z$: $G_z(t) = e^{\lambda(t-1)} \times e^{\mu(t-1)}$ — M1
$= e^{(\lambda+\mu)(t-1)} \Rightarrow X + Y \sim \text{Po}(\lambda+\mu)$ — A1 [3]
**(ii)(a)** Mean number of demands, of both types over a 2 week period is 10. — M1
$P(0 \text{ demands}) = e^{-10} = 4.54 \times 10^{-5}$ — A1 [2]
**(b)** $P(7 \text{ demands}) = P(\leq 7) - P(\leq 6) = 0.2202 - 0.1301 = 0.0901$ (using tables). — M1A1 [2]
**(iii)** Mean is now 15 — B1
$P(X \leq 26) = 0.9967$ (from tables) $\Rightarrow P(X \geq 27) = 0.0033$ — M1
Hence least $n$ is 27 — A1 [3]
**[Total: 10]**9 (i) Two independent discrete random variables $X$ and $Y$ follow Poisson distributions with means $\lambda$ and $\mu$ respectively. Prove that the discrete random variable $Z = X + Y$ follows a Poisson distribution with mean $\lambda + \mu$.
A garage has a white limousine and a green limousine for hire. Demands to hire the white limousine occur at a constant mean rate of 3 per week and demands to hire the green limousine occur at a constant mean rate of 2 per week. Demands for hire are received independently and randomly.\\
(ii) Calculate the probability that in a period of two weeks
\begin{enumerate}[label=(\alph*)]
\item no demands for hire are received, giving your answer to 3 significant figures,
\item seven demands for hire are received.\\
(iii) Find the least value of $n$ such that the probability of at least $n$ demands for hire in a period of three weeks is less than 0.005 .
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2010 Q9 [10]}}