Suppose small aircraft arrive at a certain airport at a rate of twelve per hour and that the random variable $X$ representing the number of arrivals is a Poisson random variable.
a) What is the probability that between 16 and 18 (inclusive) small aircraft arrive during a 90 min period?
b) What are the expected value and standard deviation of the number of small aircraft that arrive during a 150 minute period?
Solution
Let $X$ be the no. of aircrafts that arrive in $t$ hours. Let $\alpha = 12$ aircrafts per hour.
So $X$ follows Poisson distribution with $\lambda = \alpha t = 12 t$.
Note that 90 min = 1.5 hours. So $\lambda = \alpha * t = 12 * 1.5 = 18$.
$$ \begin{eqnarray*} P(16\leq X\leq 18) &=& \sum_{x=16}^{18}\frac{e^{-\lambda} \lambda^x}{x!} \\ &=& \frac{e^{-18}18^{16}}{16!} +\frac{e^{-18}18^{17}}{17!}+\frac{e^{-18}18^{18}}{18!}\\ &=&0.088397 + 0.093597 + 0.093597\\ &=& 0.275592. \end{eqnarray*} $$
Note that 150 min = 2.5 hours. So $\lambda = \alpha * t = 12 * 2.5 = 30$.
So the expected value of the number of small aircrafts that arrive during a 150 min period = $\lambda = 30$ aircrafts.
The standard deviation of the number of small aircrafts that arrive during a 150 min period = $\sqrt{\lambda} = \sqrt{30}=5.477$ aircrafts.
