Question

Question 3

. A city planner is considering adding a pedestrian crossing signal to a particular intersection.

They know that on average 11.5 cars go through the intersection each minute. It takes an average

person 10 seconds to walk across the intersection.

a) Name the distribution appropriate to model the number of cars that go through the intersection each

minute. Use the proper notation, and include the value of the distribution parameter(s).

b) What is the probability that a person waiting at this intersection sees no more than 2 cars in a given

minute?

c) Name the distribution appropriate to model the time to the next car crossing this intersection. Use

the proper notation, and include the value of the distribution parameter (s).

d) What is the expected value of the time to the next car crossing this intersection?

e) What is the probability that the time to the next car crossing this intersection is longer than the

average time it takes a person to traverse the intersection (that is, that the time to the next car is longer

than 10 seconds)?

f) Given your probability calculations, do you think the argument is stronger in favor or against the

construction of a pedestrian signal at this intersection? Explain your reasoning

Answer 1

a) The number of cars that go through the intersection each minute has Poisson distribution with parameter
λ 11.5

The PMF of is
P(X=x) = e-λ , x=0, 1,2, 3, xI'

b) The probability that a person waiting at this intersection sees no more than 2 cars in a given minute is

2 X! P(X 2) 0.0008

c) The inter-arrival times of the cars are exponentially distributed with rate parameter
λ 11.5
.

The PDF is $f_T\left ( t \right )=11.5e^{-11.5t};t>0$

d) The expected value of is
E (T) 1/11.5 0.0870 minutes

e) The probability,

P (T> 10/60) - 11.5e-11.5tdt 10/60 P10/60) e-11.5x10/60 P (T> 10/60) 0.1471