Question 6 - A-Level Maths

Exam Board	OCR
Module	Further Statistics AS (Further Statistics AS)
Year	2020
Session	November
Topic	Poisson Distribution
Type	Mean-variance comparison for Poisson validation

6 A statistician investigates the number, \(F\), of signal failures per week on a railway network.

The statistician assumes that signal failures occur randomly. Explain what this statement means.
State two further assumptions needed for \(F\) to be well modelled by a Poisson distribution. In a random sample of 50 weeks, the statistician finds that the mean number of failures per week is 1.61, with standard deviation 1.28.
Explain whether this suggests that \(F\) is likely to be well modelled by a Poisson distribution. Assume first that \(F \sim \operatorname { Po } ( 1.61 )\).
Write down an exact expression for \(\mathrm { P } ( F = 0 )\).
Complete the table in the Printed Answer Booklet to show the probabilities of different values of \(F\), correct to three significant figures.
Value of \(F\) 0 1 \(\geqslant 2\)
Probability 0.200
After further investigation, the statistician decides to use a different model for the distribution of \(F\). In this model it is now assumed that \(\mathrm { P } ( F = 0 )\) is still 0.200 , but that if one failure occurs, there is an increased probability that further failures occur.
Explain the effect of this assumption on the value of \(\mathrm { P } ( F = 1 )\).

Value of \(F\)	0	1	\(\geqslant 2\)
Probability	0.200