Question

Assume that the poisson distribution applies and that the mean number of hurricanes in a certain area is 7.2 per year.

Answer 1

Part A:

The poisson distribution is given by 

$P(x;\mu)=e^{-\mu}\cdot \frac{\mu^x}{x!}$

Thus, given that the mean number of hurricanes in a certain area is 7.2 per year.

The probability​ that, in a​ year, there will be 6 hurricanes is given by

$P(6;7.2)=e^{-7.2}\cdot \frac{7.2^6}{6!} \\ \\ =0.000747\times \frac{139314.0695}{720} \\ \\ =0.000747\times193.492 \\ \\ =0.1445$


Part B:

In a 45​-year ​period, the number of years that are expected to have 6 ​hurricanes is given by 0.1445 x 45 = 6.5 years.


Part C:

The poisson distribution worked here because, the observed value of the number of years that hurricanes occurred in a 45 year period which is 6 is very close to the calculated value which is 6.5