Question

1. The number of cars entering a parking lot follows Poisson distribution with mean of 4 per hour. You started a clock at some point.

a. What is the probability that you have to wait less than 30 minutes for the next car?

b. What is the probability that no car entering the lot in the first 1 hour?

c. Assume that you have wait for 20 minutes, what is the probability that you have to wait for more than 1 hour in total for the next car?

d. Find a value k such that probability that you have to wait more than k minutes is 0.4.

Answer 1

expected interarrival time β =60/4 =15 minutes

from exponential distribution:

F(x)=P(X<x)=1-e-x/β

a)

probability that you have to wait less than 30 minutes for the next car:

P(X<30)=1-exp(-30/15)=

0.8647

b)

probability that no car entering the lot in the first 1 hour(60 minutes):

P(X>60)=1-P(X<60)=1-(1-exp(-60/15))=

0.0183

c)

P(X>60 |X>20) =P(X>60)/P(X>20)=exp(-60/15)/exp(-20/15)=0.0695

d)

probability that you have to wait more than k minutes =e^-k/15 =0.4

taking long on both sides:

k=-ln(0.4)*15=13.7444 minutes