| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Topic | Poisson distribution |
| Type | Proving Poisson properties from first principles |
| Difficulty | Standard +0.3 This is a standard Further Maths question on Poisson distributions and normal approximations. Part (i) is a routine PGF derivation covered in all textbooks. Parts (ii) and (iii) are straightforward applications of normal approximation with continuity correction—standard exam technique requiring no novel insight. The calculations are mechanical once the correct approximation is identified. |
| Spec | 5.02i Poisson distribution: random events model5.02m Poisson: mean = variance = lambda5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
\begin{enumerate}[label=(\roman*)]
\item The discrete random variable $X$ has a Poisson distribution with mean $\lambda$. Use the probability generating function for $X$ to show that both the mean and the variance have the value $\lambda$. [5]
\item The number of eggs laid by a certain insect has a Poisson distribution with variance 250. Find, using a suitable approximation, the probability that between 230 and 260 (inclusive) eggs are laid. [5]
\item An insect lays 250 eggs. The probability that any egg that is laid survives to maturity is 0.1. Use a suitable approximation to find the probability that more than 30 eggs survive to maturity. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2014 Q5 [13]}}