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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2:35 minutes

Problem 5.3.9b

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.

Verified step by step guidance

Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events (e.g., births) occurring in a fixed interval of time or space, given a known average rate of occurrence (λ). The probability mass function is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).

Step 2: Calculate the average number of births per day (λ). Since there are 5942 births in 365 days, divide the total number of births by the number of days to find the daily average: λ = 5942 / 365.

Step 3: Identify the value of k. In this problem, k represents the number of births in a single day, which is given as 16.

Step 4: Substitute the values of λ and k into the Poisson probability formula. Use the formula P(X = k) = (λ^k * e^(-λ)) / k!. Replace λ with the calculated daily average and k with 16.

Step 5: Simplify the expression to compute the probability. First, calculate λ^k, e^(-λ), and k! (16 factorial). Then, divide the product of λ^k and e^(-λ) by k! to find the probability. This will give you the final result.

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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 that these events happen with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling rare events, such as the number of births in a day.

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 the given problem, λ would be calculated by dividing the total number of births (5942) by the number of days (365), which gives the expected number of births per day.

Calculating Probability with Poisson

To find the probability of observing exactly k events (in this case, 16 births) in a Poisson distribution, the formula P(X = k) = (e^(-λ) * λ^k) / k! is used, where e is the base of the natural logarithm, λ is the mean number of events, and k! is the factorial of k. This formula allows us to compute the likelihood of a specific number of occurrences given the average rate.

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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.

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.

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?

Textbook Question

Random Variable The accompanying table lists probabilities for the corresponding numbers of unlicensed software packages when four software packages are randomly selected in China. What is the random variable, what are its possible values, and are its values numerical?

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Textbook Question

In Exercises 1 and 2, determine whether the random variable x is discrete or continuous. Explain.

Let x represent the grade on an exam worth a total of 100 points.

Textbook Question

In your own words, describe the difference between the value of x in a binomial distribution and in the Poisson distribution.

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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.

Key Concepts

Poisson Distribution

Mean (λ) in Poisson Distribution

Calculating Probability with Poisson

Watch next

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.