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 $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$.\\
(i) 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 $\theta$.\\
(ii) Write down the maximum likelihood estimate of the probability that no vehicles pass during a minute.\\
(iii) 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.\\
(iv) 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

$$\frac { \theta \mathrm { e } ^ { \theta } } { \mathrm { e } ^ { \theta } - 1 } = \bar { x }$$

\hfill \mbox{\textit{OCR MEI S4 2013 Q1 [24]}}