uppose small aircr ing to a Poisso the raft arrive at a certain airport accord n

Question

uppose small aircr ing to a Poisso the raft arrive at a certain airport accord n process with rate -8 per hour, so that number of arrivals during a time period of t hours is a Poisson rv with parameter p 8t What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10? t are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period? b. Wha c. What is the probability that at least 20 small air- craft arrive during a 2.5-hour period? That at most 10 arrive during this period?

Explanation / Answer

For the Poisson distribution, P(X=k) = e- x ()k / k!

(a) For the 1-hour interval, we have = 8

The probabilities for various values of X calculated in Excel is given below:

P(X=6) = e-8 x 86 / 6! = 0.1221

P(X 6 ) = 1 - P(X=0) - P(X=1) - P(X=2) - P(X=3) - P(X=4) - P(X=5) = 1 - 0.1912 = 0.8088

P(X 10) = 1 - P (X 9) = 1 - P(X=0) - P(X=1) - P(X=2) - ... - P(X=9) = 1 - 0.7166 = 0.2834

(b) For the 90-minute duration,we have t = 1.5

So, = 1.5 x 8 = 12

E[X] = 12 aircrafts

Variance = = 12

Standard deviation = 12 = 3.4641

(c) We have t = 2.5, = 2.5 x 8 = 20

P (X 20) = 1 – P (X 19) = 1 - e-20 x [200/0! + 201 / 1! + 202 / 2! + ... + 2019 / 19!] = 1 - 0.4703 = 0.5297

P(X 10) = e-20 x [200/0! + 201 / 1! + 202 / 2! + ... + 2010 / 10!] = 0.0108

X p(X) 0 0.0003 1 0.0027 2 0.0107 3 0.0286 4 0.0573 5 0.0916 6 0.1221 7 0.1396 8 0.1396 9 0.1241 10 0.0993

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