# Probability question

• Mar 14th 2011, 10:17 PM
ulysses123
Probability question
A train bridge is constructed across a wide river. Trains arrive according to a poisson process of rate lambda=3 per day.

Find the probability that it takes more than 2 days for the 5th train to arrive at the bridge.
• Mar 15th 2011, 12:14 AM
CaptainBlack
Quote:

Originally Posted by ulysses123
A train bridge is constructed across a wide river. Trains arrive according to a poisson process of rate lambda=3 per day.

Find the probability that it takes more than 2 days for the 5th train to arrive at the bridge.

The number that arrive in a two day period has a Poission distribution with rate 6 per period.

So now your question is what is the probability of 4 or fewer trains arrive in a two day period.

CB
• Mar 15th 2011, 12:35 AM
ulysses123
in our book we have that the time of the kth arrival is a gamma distribution. I did it by evaluating the gamma distribution P(T(5)>2)=
1-P(T(5)<=2)=1-(P(T5=0)+P(T5=1)+P(T5=2)

this however gives me a difference answer to what i get doing it the way you sugested as a poison process with rate 6.
can anyone else check if the two mwthods give the same answer?
• Mar 15th 2011, 05:21 AM
CaptainBlack
Quote:

Originally Posted by ulysses123
in our book we have that the time of the kth arrival is a gamma distribution. I did it by evaluating the gamma distribution P(T(5)>2)=
1-P(T(5)<=2)=1-(P(T5=0)+P(T5=1)+P(T5=2)

this however gives me a difference answer to what i get doing it the way you sugested as a poison process with rate 6.
can anyone else check if the two mwthods give the same answer?

That will be because you have to compute:

1-P(T(5)<=2)=1-(P(T5=0)+P(T5=1|T5<>0)+P(T5=2|(T5<>0)&(T5<>1))

doing the way you suggest.

CB