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

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2:42 minutes

Problem 5.R.10b

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.

Verified step by step guidance

Step 1: Recognize that this is a Poisson probability problem. The Poisson distribution is used to model the number of events (e.g., deaths) occurring in a fixed interval of time or space, given a known average rate (λ). Here, the average rate is 7 deaths per year.

Step 2: Convert the average rate (λ) from yearly to daily. Since there are 365 days in a year, divide the yearly rate by 365 to find the daily rate: λ = 7 / 365.

Step 3: Use the Poisson probability formula to calculate the probability of 0 deaths on a given day. The formula is: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events (in this case, k = 0), λ is the average rate, and e is the base of the natural logarithm (approximately 2.718).

Step 4: Substitute k = 0 and the daily rate λ (calculated in Step 2) into the formula. Simplify the expression: P(X = 0) = (λ^0 * e^(-λ)) / 0!. Note that 0! = 1 and λ^0 = 1, so the formula simplifies to P(X = 0) = e^(-λ).

Step 5: Compute e^(-λ) using the daily rate λ. This will give you the probability of no deaths occurring on a given day.

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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 deaths in a small population, where the events occur independently of each other.

Rate Parameter (λ)

In the context of the Poisson distribution, the rate parameter (λ) represents the average number of occurrences in the specified interval. For the village of Westport, with an average of 7 deaths per year, λ would be 7. When calculating probabilities for shorter intervals, such as a day, λ must be adjusted accordingly (e.g., λ = 7/365 for daily calculations).

Probability of No Events

To find the probability of observing no events in a Poisson distribution, the formula P(X=0) = e^(-λ) is used, where e is the base of the natural logarithm. This formula indicates the likelihood of zero occurrences when the average rate is λ. In this case, it helps determine the probability of no deaths occurring on a given day in Westport.

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Related Practice

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?

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).

c. Find the probability that on a given day, there is more than one death.

Textbook Question

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

a. Find the mean number of births per day.

Textbook Question

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

b. Find the probability that in a single day, there are 16 births.

Textbook Question

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

c. Find the probability that in a single day, there are no births. Would 0 births in a single day be a significantly low number of births?

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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.

Key Concepts

Poisson Distribution

Rate Parameter (λ)

Probability of No Events

Watch next

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.