Respuesta :
Answer:
a) 0.172 probability that, in a year, there will be 4 hurricanes.
b) The expected number of years with 4 hurricanes is 7.7.
c) 7 years is close to the expected value of 7.7, 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 5.1 per year.
This means that [tex]\mu = 5.1[/tex]
a. Find the probability that, in a year, there will be 4 hurricanes.
This is P(X = 4).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 4) = \frac{e^{-5.1}*(5.1)^{4}}{(4)!} = 0.172[/tex]
0.172 probability that, in a year, there will be 4 hurricanes.
b. In a 45-year period, how many years are expected to have 4 hurricanes?
Each year, 0.172 probability of 4 hurricanes. So for 45 years, the mean is 45*0.172 = 7.7.
The expected number of years with 4 hurricanes is 7.7.
c. How does the result from part (b) compare to a recent period of 45 years in which 7 years had 4 hurricanes? Does the Poisson distribution work well here?
7 years is close to the expected value of 7.7, which means that the Poisson distribution works well here.
