6 A statistician investigates the number, $F$, of signal failures per week on a railway network.
\begin{enumerate}[label=(\alph*)]
\item The statistician assumes that signal failures occur randomly.

Explain what this statement means.
\item 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.
\item Explain whether this suggests that $F$ is likely to be well modelled by a Poisson distribution.

Assume first that $F \sim \operatorname { Po } ( 1.61 )$.
\item Write down an exact expression for $\mathrm { P } ( F = 0 )$.
\item Complete the table in the Printed Answer Booklet to show the probabilities of different values of $F$, correct to three significant figures.

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Value of $F$ & 0 & 1 & $\geqslant 2$ \\
\hline
Probability & 0.200 &  &  \\
\hline
\end{tabular}
\end{center}

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.
\item Explain the effect of this assumption on the value of $\mathrm { P } ( F = 1 )$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Statistics AS 2020 Q6 [10]}}