A group of students is discussing the conditions that are needed if a Poisson distribution is to be a good model for the number of telephone calls received by a fire brigade on a working day.
Alice says "Events must be independent". Explain why this condition may not hold in this context.
State a different condition that is needed. Explain whether it is likely to hold in this context.
The random variables \(R , S\) and \(T\) have independent Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively.
In the case \(\lambda = 2.74\), find \(\mathrm { P } ( R > 2 )\).
In the case \(\lambda = 2\) and \(\mu = 3\), find \(\mathrm { P } ( R = 0\) and \(S = 1 ) + \mathrm { P } ( R = 1\) and \(S = 0 )\). Give your answer correct to 4 decimal places.
In the general case, show algebraically that
$$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$