5 The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).
- In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context.
Assume now that \(X\) can be modelled by the distribution \(\operatorname { Po } ( 1.9 )\).
- (a) Write down an expression for \(\mathrm { P } ( X = r )\).
(b) Hence find \(\mathrm { P } ( X = 3 )\). - Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match.