| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Standard +0.3 This is a straightforward Poisson distribution question with standard normal approximation. Part (i) involves direct Poisson probability calculations with simple rate scaling. Part (ii) requires using normal approximation with continuity correction to find n, which is a routine application of the technique. All steps are standard textbook procedures with no novel insight required. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
3 The number of signal failures in a certain region of the railway network averages 10 every 3 weeks. Assume that signal failures occur independently, randomly and at constant mean rate.\\
(i) Find the probability that
\begin{enumerate}[label=(\alph*)]
\item there are between 7 and 12 (inclusive) signal failures in a three-week period,
\item there are more than 4 signal failures in a one-week period.\\
(ii) It has been calculated, using a suitable distributional approximation, that the probability of more than 62 signal failures in a period of $n$ weeks is 0.0385 . Find the value of $n$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q3}}