Answer:
a) 0.12386 = 12.386% probability that in a year, there will be 3 hurricanes.
b) 5.57 years are expected to have 3 hurricanes, that is, between 5 and 6 years.
c) 5 years is very close to the mean of 5.57, which means that the Poisson distribution works well here.
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given interval.
The mean number of hurricanes in a certain area is 5.3 per year.
This means that [tex]\mu = 5.3[/tex]
a. Find the probability that in a year, there will be 3 hurricanes.
This is P(X = 3).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = x) = \frac{e^{-5.3}*5.3^{3}}{(3)!} = 0.12386[/tex]
0.12386 = 12.386% probability that in a year, there will be 3 hurricanes.
b. in a 45-year period, how many years are expected to have 3 hurricanes ?
0.12386 each year. So, for 45 years:
45*0.12386 = 5.57
5.57 years are expected to have 3 hurricanes, that is, between 5 and 6 years.
c. How does the result from part (b) compare to a recent period of 45 years in which 5 years had 3 hurricanes? Does the Poisson distribution work well here?
5 years is very close to the mean of 5.57, which means that the Poisson distribution works well here.
