Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

Statistics

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

Previous problemNext problem

2:31 minutes

Problem 5.3.8b

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

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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.

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.

Watch next

Master Intro to Random Variables & Probability Distributions with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

Fear of Heights The table lists results from a survey of 285 subjects who were asked, “Are you afraid of heights in tall buildings?” The results are from USA Today.

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.

Textbook Question

b. In a 118-year period, how many years are expected to have no hurricanes?

Textbook Question

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

Textbook Question

Find the mean of the random variable x described in the preceding exercise.

Textbook Question

Is the mean found in the preceding exercise a statistic or a parameter?

Textbook Question

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).

a. Find the mean number of deaths per day.

Textbook Question

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).

b. Find the probability that on a given day, there are no deaths.

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

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?

Key Concepts

Poisson Distribution

Expected Value

Rate of Occurrence

Watch next

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?