Question

Changes in airport procedures require considerable planning. Arrival rates of aircrft are important factors that muct be taken into account. Suppose small aircraft arrive at a certain airport, according to a Poisson process, at the rate of 5.5 per hour(a) What is the probability that exactly 4 small aircraft arrive during a 1-hour period?(b) What is the probability that at least 4 arrive duringa 1-hour period?(c) If we define a working day as 12 hours, what isthe probability that at least 75 small aircraft arrive during a working day?

Answer 1

Answer:

a) 0.1558

b) 0.7983

c) 0.1478

Step-by-step explanation:

If we suppose that small aircraft arrive at the airport according to a Poisson process at the rate of 5.5 per hour  and if X is the random variable that measures the number of arrivals in one hour, then the probability of k arrivals in one hour is given by:

$\bf P(X=k)=\displaystyle\frac{(5.5)^ke^{-5.5}}{k!}$

(a) What is the probability that exactly 4 small aircraft arrive during a 1-hour period?

$\bf P(X=4)=\displaystyle\frac{(5.5)^4e^{-5.5}}{4!}=0.1558$

(b) What is the probability that at least 4 arrive during a 1-hour period?

$\bf P(X\geq4)=1-P(X$

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

If we redefine the time interval as 12 hours instead of one hour, then the rate changes from 5.5 per hour to 12*5.5 = 66 per working day, and the pdf is now

$\bf P(X=k)=\displaystyle\frac{(66)^ke^{-66}}{k!}$

and we want P(X ≥ 75) = 1-P(X<75). But

$\bf P(X$

hence

P(X ≥ 75) = 1-0.852 = 0.1478