Question: (a) Describe briefly one example from your own experience of a random phenomenon that could reasonably be modelled as a multivariate Poisson process. You should explain why you think the model is appropriate.

(b) During normal working hours, the help desk of a software company receives telephone calls according to a Poisson process at an average rate of three calls every twenty minutes.

1. Find the probability that during a ten-minute period

no call is received;

at least two calls are received.

From experience it has been noted that $25 \%$ of calls relate to the company's spreadsheet package.

(iii) Find \(\lambda_{s}\), the hourly arrival rate of calls relating to the spreadsheet package, and stale the distribution of the number of calls relating to the spreadsheet package that are received in two hours. Calculate the probability that more than two such calls are received in a two-hour period.

(iv) Calculate the probability that exactly five calls are received between 11 am and 11.20 am, exactly two of which relate to the spreadsheet package.

(v) Exactly eight calls are received between 11 am and 11.20 am one morning. Calculate the probability that exactly three of them relate to the spreadsheet package.

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