Question

State and local governments conduct traffic studies for a variety of reasons. Studies can be components of broader economic impact studies to understand how a new proposed development might impact existing traffic flows. Such studies can also be focused on a specific question such as whether additional traffic control (lights and signage) are needed at a particular intersection, or if additional lanes are needed on a busy road to relieve traffic congestion. State transportation departments also use traffic studies to understand the flow of vehicles on highways.

You probably have been a participant in a traffic study at some point. If you have ever driven over mysterious looking cables lying in the road, you and your vehicle became an observation!. These cables are connected to a box that increments the count as vehicles drive over them (if multiple cables are present, your speed might also be measured).

The Poisson distribution is often used in traffic studies to model traffic flow. In this context, an "arrival" is the passage of a vehicle over some fixed point in the road, such as an intersection. Suppose that traffic passes through a particular intersection in a city at an average rate of 48.7 vehicles every 15 minutes. What is the probability that over 100 vehicles will pass though the intersection in any 30-minute period, to three decimal places?

Hint: I strongly suggest you use software such as Excel; it will save you a lot of calculations! Also, be sure to use the correct value of λ.

Please help with my business analysis 2372 homework i missed. I need to know how to work it out.

Answer 1

The average rate traffic passes the intersection is 48.7 vehicles per 15 minutes       
Hence, in 30 minutes the average traffic passing is 48.7*2 = 97.4 vehicles per 30 minutes       
Let X be the number of vehicles passing in 30 minutes       
X follows a Poisson distribution with λ = 97.4 vehicles in 30 minutes       
The pdf of the Poisson distribution is        

That is        
       
        
a) To find P(over 100 vehicles will pass though the intersection in any 30-minute period)       
that is to find P(X > 100)       
P(X > 100) = 1- P(X ≤ 100)       
We use the Excel function POISSON.DIST to find the probability       
P(X > 100) = 1 - POISSON.DIST(100, 97.4, TRUE)       
                 = 1 - 0.6290       
                 = 0.3710       
P(over 100 vehicles will pass though the intersection in any 30-minute period) = 0.3710