Question

The sutton police department must write, on average, 6 tickets a day to keep department revenues at budgeted levels. suppose the number of tickets written per day follows a poisson distribution with a mean of 6.5 tickets per day. interpret the value of the mean.

Answer 1

The Poisson distribution defines the probability of k discrete and independent events occurring in a given time interval.
If λ = the average number of event occurring within the given interval, then

`P(k \, events) = e^(-\lambda ) ( (\lambda ^(k))/(k!) )`

For the given problem,
λ = 6.5, average number of tickets per day.
k = 6, the required number of tickets per day
The Poisson distribution is

`P(k \, tickets/day)=e^(-6.5) ( (6.5 ^(k))/(k!) )`
The distribution is graphed as shown below.

Answer:
The mean is λ = 6.5 tickets per day, and it represents the expected number of tickets written per day.
The required value of k = 6 is less than the expected value, therefore the department's revenue target is met on an average basis.