Answer to Question #347214 in Statistics and Probability for shiba

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

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

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

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

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