591 Traffic model The number of vehicles passing a particula

5.91 Traffic model. The number of vehicles passing a particular mile marker during 15-minute units of time can be modeled as a Poisson random variable. Counting devices show that the average number of vehicles passing the mile marker every 15 minutes is 48.7.

(a) What is the probability of 50 or more vehicles passing the marker during a 15-minute time period?

(b) What is the standard deviation of the number of vehicles passing the marker in a 15-minute time period? A 30-minute time period?

(c) What is the probability of 100 or more vehicles passing the marker during a 30-minute time period?

Solution

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
vehicles passing the mile marker every 15 minutes is 48.7
=48.7
a)
USED AN EXCCEL FORMULA : ROUND(POISSON(50,48.7,TRUE),4)
P (X < 50) = 0.555
P( X > = 50 ) = 1 - P (X < 50) = 0.445

b)
For 15 mins s.d = Sqrt() = Sqrt(48.7)= 6.9785
For 30 mins s.d = Sqrt() = Sqrt(48.7*2)= 9.8691

c)
USED AN EXCCEL FORMULA : ROUND(POISSON(100,97.4,TRUE),4)
P (X < 100) = 0.5905
P( X > = 100 ) = 1 - P (X < 100) = 0.4095