In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

Hurricanes

b. In a 118-year period, how many years are expected to have 7 hurricanes?

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Understand the problem: The Poisson distribution is used to model the number of events (hurricanes) occurring in a fixed interval of time (years). The mean number of hurricanes per year is given as 5.5. We are tasked with finding the expected number of years with exactly 7 hurricanes over a 118-year period.

Recall the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the number of events we are interested in, and e is the base of the natural logarithm (approximately 2.718).

For a single year, calculate the probability of exactly 7 hurricanes using the Poisson PMF formula. Substitute λ = 5.5 and k = 7 into the formula: P(X = 7) = (5.5^7 * e^(-5.5)) / 7!.

To find the expected number of years with 7 hurricanes over a 118-year period, multiply the probability of 7 hurricanes in a single year by the total number of years: Expected years = P(X = 7) * 118.

Simplify the expression to compute the final result. This involves calculating the factorial of 7 (7!), raising 5.5 to the power of 7, and multiplying by e^(-5.5), then multiplying the result by 118.

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주요 개념

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Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling rare events, such as the number of hurricanes in a year, where the events are independent of each other.

Expected Value

The expected value is a key concept in probability that represents the average outcome of a random variable over a large number of trials. In the context of the Poisson distribution, the expected number of occurrences can be calculated by multiplying the mean rate of occurrence by the number of intervals considered, providing a way to predict outcomes over time.

Rate of Occurrence

The rate of occurrence in a Poisson distribution refers to the average number of events (in this case, hurricanes) expected to happen in a specified time frame. For the given problem, the mean number of hurricanes is 5.5 per year, which serves as the basis for calculating probabilities and expected occurrences over longer periods, such as 118 years.