Statistics question?

Question

On average there are 2.2 planes arriving at a local airport in an hour daily.

1.What is the probability that there are 3 planes arriving at the airport in an hour?

2.What is the probability that there are at least 2 planes arriving at the airport in a two-hour?

3.What is the standard deviation for the number of planed arriving at the airport in a 10 hour period?

Answer 1

It entirely depends on the type of distribution you assume the arrivals follow.

For instance if the arrivals are uniformly distributed then the answers will be very different than if you choose, say, a poisson distribution, etc...

Furthermore, since you are talking about scheduled airlines flights, a uniform distribution is entirely reasonable to assume; generally a Poisson would be usually considered for random arrivals and events.

Other distributions can be useful to - for instance if the flights density is greater during daytime hours than night time then a Weibull distribution can be useful.

Answer 2

It is reasonable to assume that the arrivals can be modelled by a Poisson process with distribution Poisson(m), where m is the mean number of arrivals in a given number of hours. For this distribution, the probability of a particular number of arrivals, x, is p(x) = (e^(-m)m^x) / x!

(1) For a one-hour period, the distribution is Poisson(2.2).
p(3) = (e^(-2.2)2.2^3) / 3! = 0.1966

(2) For a two-hour period, the distribution is Poisson(4.4).
Probability of at least two planes in this period is = 1 – (p(0) + p(1)).
p(0) = (e^(-4.4)4.4^0) / 0! = 0.0123
p(1) = (e^(-4.4)4.4^1) / 1! = 0.0540
So probability of at least two planes in two hours = 1 – 0.0663 = 0.9337.

(3) For Poisson(m) the standard deviation is sqrt(m). For a 10-hour period the distribution is Poisson(22) so the standard deviation is sqrt(22) = 4.6904.