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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.T.4
Chapter 4, Problem 4.T.4

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.

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Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, given the average number of occurrences (denoted as μ). Here, μ = 5 represents the mean number of customer arrivals per minute.
Step 2: Generate 20 random numbers using the Poisson distribution with μ = 5. This can be done using statistical software or programming languages such as Python (using numpy's poisson function), R, or Excel. Each number represents the number of arrivals in one minute.
Step 3: Create a table to track the number of customers waiting at the end of each minute. For each minute, compare the number of arrivals (from the Poisson distribution) to the store's processing capacity (4 customers per minute). If arrivals exceed the processing capacity, calculate the number of customers left waiting and carry them over to the next minute.
Step 4: Update the table iteratively for all 20 minutes. For each minute, add the number of new arrivals to the number of customers carried over from the previous minute, subtract the processing capacity (4 customers), and record the remaining customers waiting.
Step 5: At the end of 20 minutes, review the table to determine the total number of customers waiting. The table should include columns for minute, arrivals, processed customers, and customers waiting.

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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 times an event happens in a specified period, such as customer arrivals at a store. The parameter 'mu' represents the average rate, which in this case is set to 5 arrivals per minute.
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Random Number Generation

Random number generation is the process of creating a sequence of numbers that cannot be reasonably predicted better than by random chance. In the context of the Poisson distribution, random numbers can be generated to simulate customer arrivals, allowing for the analysis of waiting times and service efficiency. This technique is essential for creating realistic models in statistical simulations.
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Queueing Theory

Queueing theory is the mathematical study of waiting lines or queues. It helps analyze the behavior of queues in various systems, such as customer service in a grocery store. By understanding the arrival rate and service rate, one can predict the number of customers waiting at any given time, which is crucial for managing resources and improving service efficiency.
Related Practice
Textbook Question

Determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

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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

c. one customer is waiting in line after one minute and no customers are waiting in line after the second minute..

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Textbook Question

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (a) one or two HD televisions

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Textbook Question

The table shows the ages of students in a freshman orientation course.

a. Construct a probability distribution.

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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 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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