Textbook Question
Determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.
Ch. 4 - Discrete Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 4, Problem 4.T.7c
In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.
The mean number of arrivals per minute is four. Find the probability that
c. one customer is waiting in line after one minute and no customers are waiting in line after the second minute..
Verified step by step guidance1
Step 1: Recognize that this problem involves a Poisson process, as the mean number of arrivals per minute is given (λ = 4), and we are dealing with the probability of a specific number of arrivals in a fixed time interval.
Step 2: Use the Poisson probability formula: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of arrivals per minute, k is the number of arrivals, and e is the base of the natural logarithm.
Step 3: For the first minute, calculate the probability that one customer is waiting in line. This means there were 5 arrivals (4 processed + 1 waiting). Use the Poisson formula with k = 5 and λ = 4.
Step 4: For the second minute, calculate the probability that no customers are waiting in line. This means the number of arrivals equals the number of customers processed (4). Use the Poisson formula with k = 4 and λ = 4.
Step 5: Multiply the probabilities from Step 3 and Step 4 to find the joint probability that one customer is waiting after the first minute and no customers are waiting after the second minute.
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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 the number of arrivals in a queue, such as customers at a grocery store, where events occur independently.
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Intro to Frequency Distributions
Queueing Theory
Queueing theory is the mathematical study of waiting lines or queues. It helps analyze the behavior of queues in terms of arrival rates, service rates, and the number of servers. In this context, it can be used to determine the likelihood of a certain number of customers waiting in line at a grocery store checkout.
Probability Calculation
Probability calculation involves determining the likelihood of a specific event occurring, often expressed as a number between 0 and 1. In this scenario, it requires calculating the probabilities of having one customer waiting after one minute and no customers waiting after the second minute, using the Poisson distribution and the principles of queueing theory.
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Related Practice
Textbook Question
In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.
The mean increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for mu = 5 . Then create a table that shows the number of customers waiting at the end of 20 minutes.
Textbook Question
In Exercises 1–3, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (a) the first return requiring an audit is the 25th return the tax auditor examines, (b) the first return requiring an audit is the first or second return the tax auditor examines, and (c) none of the first five returns the tax auditor examines require an audit. (Source: Kiplinger)
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Textbook Question
In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.
The mean number of arrivals per minute is four. Find the probability that
a. three, four, or five customers will arrive during the third minute.
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Textbook Question
The table shows the ages of students in a freshman orientation course.
a. Construct a probability distribution.
Textbook Question
In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.
The mean number of customers who arrive at the checkout counters each minute is 4. Create a Poisson distribution with mu = 4 for x = 0 to 20. Compare your results with the histogram shown at the upper right.
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