# Poisson question

• Jul 19th 2011, 11:53 PM
downthesun01
Poisson question
Changes in airport procedures require considerable planning. Arrival rates of aircraft are important factors that must be taken into account. Suppose small aircraft arrive at a certain airport, according to a Poisson process, at the rate of 6 per hour. Thus, the Poisson parameter for arrivals over a period of hours is μ = 6t.

If we define a working day as 12 hours, what is the probability that at least 75 small aircraft arrive during a working day?

Ok, the new $\lambda =72$

But how do I find $pr(X\geq 75)$?

I know that $pr(X\geq 75)=1-pr(X\leq 74)$

but it seems very time consuming to have to find $pr(X=0), pr(X=1),...,pr(X=73), pr(X=74)$

Is there a faster way of solving the problem?
• Jul 20th 2011, 01:54 AM
Siron
Re: Poisson question
Do you have a calculator? ...
• Jul 20th 2011, 03:07 AM
downthesun01
Re: Poisson question
I went ahead and used Excel to solve the problem, but if there's some sort of shortcut or something that allows for the answer to be found quickly by hand I'd love to know it. Thanks
• Jul 20th 2011, 03:08 AM
CaptainBlack
Re: Poisson question
Quote:

Originally Posted by downthesun01
Changes in airport procedures require considerable planning. Arrival rates of aircraft are important factors that must be taken into account. Suppose small aircraft arrive at a certain airport, according to a Poisson process, at the rate of 6 per hour. Thus, the Poisson parameter for arrivals over a period of hours is μ = 6t.

If we define a working day as 12 hours, what is the probability that at least 75 small aircraft arrive during a working day?

Ok, the new $\lambda =72$

But how do I find $pr(X\geq 75)$?

I know that $pr(X\geq 75)=1-pr(X\leq 74)$

but it seems very time consuming to have to find $pr(X=0), pr(X=1),...,pr(X=73), pr(X=74)$

Is there a faster way of solving the problem?

The probability that at least 75 aircraft arriving is indeed one minus the probability of 74 or fewer arriving, so you would need to calculate:

$Pr(X\le 74)= Pr(X=0)+Pr(X=1)+ ... + Pr(X=74)$

which is too tedious to compute by hand, so you will probably want to use the normal approximation if you have been shown it.

CB