Question

A category 4 hurricane strikes the U. S. mainland once every 4 years, on average,a) What are the expected value and standard deviation of the number of category 4 hurricanes that will strike the U. S. mainland in the next century (100 years)?b) What is the probability that exactly 3 such hurricanes will strike the U. S. mainland in the next 10 years?

Answer 1

a) The mean is 25 and the standard deviation is 5.

b) 21.38% probability that exactly 3 such hurricanes will strike the U. S. mainland in the next 10 years

Explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

In which

x is the number of sucesses

e = 2.71828 is the Euler number

`\mu` is the mean in the given interval. The variance is the same as the mean.

a) What are the expected value and standard deviation of the number of category 4 hurricanes that will strike the U. S. mainland in the next century (100 years)?

Mean of 1 category 4 hurricane each 4 years. So for 100 years.

`\mu = (100*1)/(4) = 25`

Variance is the same as the mean, so 25.

The standard deviation is the square root of the variance. So it is the square root of 25, which is 5.

b) What is the probability that exactly 3 such hurricanes will strike the U. S. mainland in the next 10 years?

Mean of 1 category 4 hurricane each 4 years. So for 10 years.

`\mu = (10*1)/(4) = 2.5`

This probability is P(X = 3). So

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

`P(X = 3) = (e^(-2.5)*(2.5)^(3))/((3)!) = 0.2138`

21.38% probability that exactly 3 such hurricanes will strike the U. S. mainland in the next 10 years