Poisson Distribution Question

[Dax]

Hey Guys, could someone tell me how to do the Poisson Distribution Question in Section A of Paper 2 TZ1 2013 May?

Thanks!

[Negotiation]

Doesn't the mark scheme lay it out nicely? It's formatted exactly the way my answer is. What are you finding tricky?

Edited April 26, 2014 by Negotiation

[rinik]

a)

In T minutes there will be T/4 cars and hence the parameter you use is 0.25T

P(X<=3)= P(0)+P(1)+P(2)+P(3)=0.6

Substitute 0.25T as the parameter in the equation and solve using a gdc.

for example P(0)= (0.25T)^0*e^(-0.25T)/0!

do this for all of the probabilities and you get an equation with one unknown, T

b)

A ferry can carry a max of 3 cars. That means that a total of 6 can arrive because if more arrive than it will take another 10 minutes to transport all of the cars. This means that at least 3 have to arrive in the first 10 minutes because if 4 or more cars arrive in the second 10 minutes you will need another 10 to transport all of the cars. The new parameter is 10/4=2.5

There are 4 possibilities

No more than 3 cars arrive in the first 10 minutes and no more than 3 cars arrive in the next 10 minutes

4 cars arrive in the first 10 min and no more than 2 arrive in the next 10

5 cars arrive in the first 10 minutes and no more arrive than 1 in the next 10

6 arrive in the first 10 minutes and no more than 0 arrive in the next 10 minutes.

no more than= cumulative distribution

Just multiply the values for each situation and add up all 4 situations and you get the required probability. Use poissonpdf and poissoncdf or this will take you forever.

Edit:

you can also do this with a lot longer method. Using the same parameter and the fact that there can be a max of 6 cars you can calculate the probabilities individually for everything and sum them up

P(0)P(0)+P(1)P(0)+P(2)P(0)+P(3)P(0)+P(3)P(1)..............................P(6)P(0).

Try expanding the cumulative probabilities in the previous method and you will see that both expressions are the same

I am writing this to show you that the probabilities when less than 6 cars arrive are present in the first method

Edited April 27, 2014 by rinik