| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Topic | Approximating the Poisson to the Normal distribution |
| Type | Combined Poisson approximation and exact calculation |
| Difficulty | Moderate -0.3 This is a straightforward application of standard Poisson-to-Normal approximation techniques. Part (i) requires calculator/tables for cumulative Poisson probability, part (ii) applies the continuity correction and standardization formula (routine for Further Maths students), and part (iii) asks for a standard comment about approximation validity. While it involves multiple parts, each step follows textbook procedures without requiring problem-solving insight or novel approaches, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
(i) $\text{P}(X \leq 5) = e^{-12.25}\left(1 + 12.25 + \dfrac{12.25^2}{2!} + \dfrac{12.25^3}{3!} + \dfrac{12.25^4}{4!} + \dfrac{12.25^5}{5!}\right)$
$= 0.0174$ **M1A1, A1** [3]
(ii) $z = \dfrac{5.5 - 12.25}{3.5} = -1.929$ **M1A1**
$\text{P}(X \leq 5)$ (from tables) $= 1 - 0.9731 = 0.0269$ (Allow 0.027(0)) **A1** [3]
(iii) Result is not reliable (error is approximately 50%). **B1**
The mean is not large enough. **B1** [2]2 The discrete random variable $X$ has a Poisson distribution with mean 12.25.\\
(i) Calculate $\mathrm { P } ( X \leqslant 5 )$.\\
(ii) Calculate an approximate value for $\mathrm { P } ( X \leqslant 5 )$ using a normal approximation to the Poisson distribution.\\
(iii) Comment, giving a reason, on the accuracy of using a normal approximation to the Poisson distribution in this case.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2011 Q2 [8]}}