Questions: Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 7.2 p a. Find the probability that, in a year, there will be 5 hurricanes. b. In a 45 -year period, how many years are expected to have 5 hurricanes? c. How does the result from part (b) compare to a recent period of 45 years in which 5 years had 5 hurricanes? Poisson distribution work well here?

Solution Steps

To find the probability of observing exactly \( k = 5 \) hurricanes in a year when the mean number of hurricanes \( \lambda = 7.2 \), we use the Poisson probability mass function:

\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]

Substituting the values:

\[ P(X = 5) = \frac{7.2^5 e^{-7.2}}{5!} \approx 0.1204 \]

Thus, the probability of having 5 hurricanes in a year is:

\[ \text{Probability of having 5 hurricanes in a year} \approx 0.1204 \]

In a 45-year period, the expected number of years with 5 hurricanes can be calculated by multiplying the probability from Step 1 by the number of years:

\[ \text{Expected years} = P(X = 5) \times 45 \approx 0.1204 \times 45 \approx 5.418 \]

Thus, the expected number of years with 5 hurricanes in 45 years is approximately:

\[ \text{Expected number of years with 5 hurricanes} \approx 5.418 \]

In a recent 45-year period, it was observed that there were 5 years with 5 hurricanes. We compare this observed value with the expected value calculated in Step 2:

\[ \text{Observed years} = 5 \]

Since \( 5.418 \) (expected) does not equal \( 5 \) (observed), we conclude that:

\[ \text{The Poisson distribution model does not match the observed data well.} \]

Final Answer