1 Traffic engineers are studying the flow of vehicles along a road. At an initial stage of the investigation, they assume that the average flow remains the same throughout the working day. An automatic counter records the number of vehicles passing a certain point per minute during the working day. A random sample of these records is selected; the sample values are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\).

1. The engineers model the underlying random variable \(X\) by a Poisson distribution with unknown parameter \(\theta\). Obtain the likelihood of \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\) and hence find the maximum likelihood estimate of \(\theta\).
2. Write down the maximum likelihood estimate of the probability that no vehicles pass during a minute.
3. The engineers note that, in a sample of size 1000 with sample mean \(\bar { x } = 5\), there are no observations of zero. Suggest why this might cast some doubt on the investigation.
4. On checking the automatic counter, the engineers find that, due to a fault, no record at all is made if no vehicle passes in a minute. They therefore model \(X\) as a Poisson random variable, again with an unknown parameter \(\theta\), except that the value \(x = 0\) cannot occur. Show that, under this model, $$\mathrm { P } ( X = x ) = \frac { \theta ^ { x } } { \left( \mathrm { e } ^ { \theta } - 1 \right) x ! } , \quad x = 1,2 , \ldots$$ and hence show that the maximum likelihood estimate of \(\theta\) satisfies the equation $$\frac { \theta \mathrm { e } ^ { \theta } } { \mathrm { e } ^ { \theta } - 1 } = \bar { x }$$