If arrivals are Poisson distributed 15 per hour answer the f

If arrivals are Poisson distributed 15 per hour, answer the following. [Hint: Determine the mean time between arrivals and use the distribution that addresses time between events]

8.1   Probability that the time between two consecutive arrivals will be between 3.5 and 4.5 minutes

8.2   Probability that the time between consecutive arrivals on three of the next four occasions will be between 3.5 and 4.5 minutes

8.3   Probability that the time between two consecutive arrivals will be exactly 4 minutes

8.4   Maximum time between consecutive arrivals

8.5   Distribution’s mean and standard deviation

Solution

If no of arrivals is Poisson with 15 per hour

then mean time between arrivals = 60/15 = 4 minutes

The waiting time follow exponential with mean = 4 minutes

8.1.P(3.5<x<4.5)=0.0461

8.2 3 of next 4 hours. No of hours binomial Y

Y is binomial with p = 0.0461 as per 8.2 and n = 4

P(Y=3) = 0.00038

8.3 P(X=4) = 0.1024 (exactly 15 customers arrive)

8.4 Maximum time

8.5 Distribution mean and sigma

Mean = 4

std dev = 4