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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
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Tutti i libri di testoTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblema 5.3.7b
Capitolo 5, Problema 5.3.7b

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 no hurricanes?

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1
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 no hurricanes over a 118-year period.
Step 1: 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, and e is the base of the natural logarithm (approximately 2.718).
Step 2: For this problem, we are interested in the probability of having no hurricanes in a single year (k = 0). Substitute k = 0 and λ = 5.5 into the PMF formula: P(X = 0) = (5.5^0 * e^(-5.5)) / 0!. Simplify this expression, noting that 0! = 1 and 5.5^0 = 1.
Step 3: Once you calculate P(X = 0), interpret it as the probability of having no hurricanes in a single year. To find the expected number of years with no hurricanes over a 118-year period, multiply this probability by the total number of years: Expected years = P(X = 0) * 118.
Step 4: Perform the calculations step by step to determine the final expected number of years with no hurricanes. Ensure that you use accurate values for e^(-5.5) and complete the multiplication with 118 to arrive at the result.

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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 natural disasters, where the events are independent of each other.
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Intro to Frequency Distributions

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 average rate (mean) by the number of intervals considered, providing a basis for predicting outcomes.
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Expected Value (Mean) of Random Variables

Probability of Zero Events

In a Poisson distribution, the probability of observing zero events in a given interval can be calculated using the formula P(X=0) = e^(-λ), where λ is the mean number of events. This concept is crucial for determining how many years in a specified period are expected to have no hurricanes, as it directly relates to the average rate of occurrence.
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Probability of Multiple Independent Events
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