6. Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5 t\).
- Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m.
- A customer arrives at exactly 11 a.m.
- Let the next customer arrive at \(T\) minutes past 11 a.m. Show that
$$P ( T > t ) = \mathrm { e } ^ { - 0.5 t }$$
- Hence find the probability density function, \(f ( t )\), of \(T\).
- Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers.