Question

2. 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 5 per hour. Thus, the Poisson parameter for arrivals over a period of hours is μ = 5t.

(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 during a 1-hour period?

(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?

Answer 1

a)

$$P(X=4)=\dfrac{e^{-5}(5)^4}{4!}=0.175467$$

b)

$$P(X\ge4)=1-P(X=0)-P(X=1)$$

$$-P(X=2)-P(X=3)$$

$$=1-\dfrac{e^{-5}(5)^0}{0!}-\dfrac{e^{-5}(5)^1}{1!}$$

$$-\dfrac{e^{-5}(5)^2}{2!}-\dfrac{e^{-5}(5)^3}{3!}=0.734974$$

c)

$$\lambda t=5(12)=60$$

The Poisson distribution can be approximated with Normal when λ is large

$\mu=\lambda t=60, \sigma^2=\lambda t=60$

$$P(X\ge75)\approx P(X>74.5)$$

$$=1-P(Z\le\dfrac{74.5-60}{\sqrt{60}})$$

$$=1-P(Z\le1.871942)\approx0.0306$$