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.
b. In a 118-year period, how many years are expected to have 10 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 \( \lambda = 5.5 \). We are tasked with finding the expected number of years with 10 hurricanes over a 118-year period.

Recall the formula for the Poisson probability mass function (PMF): \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \), where \( k \) is the number of events (hurricanes), \( \lambda \) is the mean number of events, and \( e \) is the base of the natural logarithm.

Calculate the probability of exactly 10 hurricanes in a single year using the Poisson PMF formula. Substitute \( \lambda = 5.5 \) and \( k = 10 \) into the formula: \( P(X = 10) = \frac{e^{-5.5} (5.5)^{10}}{10!} \). Simplify the expression but do not compute the final value.

To find the expected number of years with 10 hurricanes over a 118-year period, multiply the probability of 10 hurricanes in a single year by the total number of years: \( \text{Expected years} = P(X = 10) \times 118 \).

Substitute the previously calculated \( P(X = 10) \) into the formula \( \text{Expected years} = P(X = 10) \times 118 \). Simplify the expression but do not compute the final value.

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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 value is equal to the mean (λ), which indicates the average number of occurrences—in this case, the average number of hurricanes per year.

Rate of Occurrence

The rate of occurrence in a Poisson distribution refers to the average number of events (hurricanes) expected to happen in a specified time frame (one year). In this scenario, with a mean of 5.5 hurricanes per year, this rate helps determine the likelihood of observing a specific number of hurricanes over a longer period, such as 118 years.

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.b. In a 118-year period, how many years are expected to have 10 hurricanes?