In Exercises 5–8, assume that the Poisson distribution applies; a...

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

Statistics

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

1:52 minutes

Problem 5.3.8a

Textbook Question

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.
a. Find the probability that in a year, there will be 10 hurricanes.

Verified step by step guidance

Understand the problem: The Poisson distribution is used to model the number of events (hurricanes) occurring in a fixed interval of time (a year) when the events occur independently and at a constant average rate. Here, the mean number of hurricanes per year (λ) is 5.5, and we are tasked with finding the probability of observing exactly 10 hurricanes in a year.

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).

Substitute the given values into the formula: Set λ = 5.5 and k = 10. The formula becomes P(X = 10) = ((5.5)^10 * e^(-5.5)) / 10!.

Break down the calculation into manageable parts: (1) Compute (5.5)^10, which is the mean raised to the power of 10. (2) Compute e^(-5.5), which is the exponential function with a negative mean. (3) Compute 10!, which is the factorial of 10 (i.e., 10 × 9 × 8 × ... × 1).

Combine the results: Multiply (5.5)^10 by e^(-5.5), then divide the result by 10!. This will give you the probability of observing exactly 10 hurricanes in a year.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the specified interval. For this question, λ is given as 5.5, indicating that, on average, there are 5.5 hurricanes per year in the United States, which serves as a key parameter for calculating probabilities.

Calculating Probability with Poisson Formula

To find the probability of observing a specific number of events (k) in a Poisson distribution, the formula P(X = k) = (e^(-λ) * λ^k) / k! is used, where e is Euler's number (approximately 2.71828), λ is the mean, and k! is the factorial of k. This formula allows us to compute the likelihood of experiencing exactly k events, such as 10 hurricanes in a year.

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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.a. Find the probability that in a year, there will be 10 hurricanes.