Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

Statistics

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

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2:05 minutes

Problem 5.3.7a

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 no hurricanes.

Verified step by step guidance

Step 1: Recall the formula for the Poisson probability distribution: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of occurrences, k is the number of occurrences we are interested in, and e is the base of the natural logarithm (approximately 2.718).

Step 2: Identify the given values from the problem. Here, λ = 5.5 (the mean number of hurricanes per year) and k = 0 (since we are finding the probability of no hurricanes in a year).

Step 3: Substitute the values into the Poisson formula. This gives P(X = 0) = (5.5^0 * e^(-5.5)) / 0!.

Step 4: Simplify the terms in the formula. Note that any number raised to the power of 0 is 1, so 5.5^0 = 1. Also, 0! (zero factorial) is equal to 1. Therefore, the formula simplifies to P(X = 0) = (1 * e^(-5.5)) / 1.

Step 5: The final step is to compute e^(-5.5) and divide by 1 to find the probability. This step involves using a calculator or software to evaluate the exponential term.

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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.

Calculating Probability with Poisson

To find the probability of observing a specific number of events in a Poisson distribution, the formula P(X=k) = (e^(-λ) * λ^k) / k! is used, where P(X=k) is the probability of k events occurring, e is Euler's number, and k! is the factorial of k. For this question, to find the probability of no hurricanes (k=0), you would substitute λ=5.5 and k=0 into the formula.

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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.

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Textbook Question

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