| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2026 |
| Session | January |
| Marks | 6 |
| Topic | Poisson distribution |
| Type | Expectation and variance of Poisson-related expressions |
| Difficulty | Standard +0.3 This question tests standard properties of Poisson distributions (additivity and variance rules) with straightforward application of normal approximation. Part (i)(a) requires recognizing X+Y ~ Po(31) and using normal approximation with continuity correction. Part (i)(b) is direct application of Var(aX+bY) = a²Var(X) + b²Var(Y). Part (ii) asks for the independence assumption. All steps are routine for Further Maths Statistics students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson |
4.
The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by $X$ and $Y$ respectively. It is given that $X \sim \operatorname { Po } ( 17 )$ and $Y \sim \operatorname { Po } ( 14 )$.\\
(i) Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( X + Y ) > 40$,
\item $\operatorname { Var } ( 2 X - Y )$.\\
(ii) State a necessary assumption for your calculations in part (i) to be valid.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2026 Q4 [6]}}