4 The numbers of call-outs per day received by a fire station for a random sample of 255 weekdays were recorded as follows. The mean number of call-outs per day for these data is 0.6 . A Poisson model, using this sample mean of 0.6 , is fitted to the data, and gives the following expected frequencies (correct to 3 decimal places).

1. Using a \(5 \%\) significance level, carry out a test to examine the goodness of fit of the model to the data. The time \(T\), measured in days, that elapses between successive call-outs can be modelled using the exponential distribution for which \(\mathrm { f } ( t )\), the probability density function, is $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 ,
   \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 , \end{cases}$$ where \(\lambda\) is a positive constant.
2. For the distribution above, it can be shown that \(\mathrm { E } ( T ) = \frac { 1 } { \lambda }\). Given that the mean time between successive call-outs is \(\frac { 5 } { 3 }\) days, write down the value of \(\lambda\).
3. Find \(\mathrm { F } ( t )\), the cumulative distribution function.
4. Find the probability that the time between successive call-outs is more than 1 day.
5. Find the median time that elapses between successive call-outs.