1st year probability distribution question

Subhash · #1 $Old$ August 26th, 2008, 08:17 AM

Hey,

I was stuck on this question, and have no idea how to go on about solving this. If some one can guide me in this question, it would be really appreciated.

Aeroplanes arrive at an airport at the average of 5.5 per minute. Traffic controllers can safely handle a maximum of 7 planes per min. What is the probability that airport is safety is jepordised? Planes arrive randomly and independently.

Thanks

Moo · #2 $Old$ August 26th, 2008, 08:24 AM

Hello,

Quote:

Originally Posted by Subhash $View Post$

Hey,

I was stuck on this question, and have no idea how to go on about solving this. If some one can guide me in this question, it would be really appreciated.

Aeroplanes arrive at an airport at the average of 5.5 per minute. Traffic controllers can safely handle a maximum of 7 planes per min. What is the probability that airport is safety is jepordised? Planes arrive randomly and independently.

Thanks

Isn't it a Poisson distribution ?

Quote:

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event.

Poisson distribution - Wikipedia, the free encyclopedia

X : number of planes
$X \leadsto \mathcal{P}(\lambda)$ , with $\lambda=5.5$

drichie · #3 $Old$ August 27th, 2008, 02:22 PM

Quote:

Originally Posted by Subhash $View Post$

Hey,

I was stuck on this question, and have no idea how to go on about solving this. If some one can guide me in this question, it would be really appreciated.

Aeroplanes arrive at an airport at the average of 5.5 per minute. Traffic controllers can safely handle a maximum of 7 planes per min. What is the probability that airport is safety is jepordised? Planes arrive randomly and independently.

Thanks

I think so.

I am assuming the inter-arrival time follows exponential distribution.ie I am assuming the number of planes arrive per minute follows poisson with mean (lamda) = 5.5

So. probability that airport is safety is jepordised
= P[X>8] = 1 - P[X<=7]
= 1- { exp(-5.5) ( 1 + 5.5/1 + 5.5^2/2! +5.5^3/3! +5.5^4/4! +5.5^5/5! +5.5^6/6! +5.5^7/7! )}

= 1- 0.8094853
=0.19