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Chapter 5: Problem 2

When using the Poisson distribution, which parameter of the distribution is used in probability computations? What is the symbol used for this parameter?

Short Answer

Expert verified
The parameter is \( \lambda \), representing the average rate of events.

Step by step solution

01

Understanding the Poisson Distribution

The Poisson distribution is a probability distribution used to model the number of events occurring within a fixed interval of time or space. It is ideal for scenarios where events happen independently of each other.
02

Identifying the Parameter

In the Poisson distribution, the key parameter is the average rate at which events occur in a given time period, denoted as \( \lambda \). This parameter represents both the mean and the variance of the distribution.
03

Using the Parameter in Probability Computations

The parameter \( \lambda \) is used extensively in the probability mass function of the Poisson distribution. The formula to find the probability of observing \( k \) events is given by: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]where \( e \) is the base of the natural logarithm, \( k \) is the number of events, and \( k! \) is the factorial of \( k \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Mass Function
The Probability Mass Function (PMF) is a crucial concept when dealing with the Poisson distribution. It provides the probability that a discrete random variable is exactly equal to some value.
For a Poisson distribution, this function calculates the likelihood of a certain number of events occurring in a fixed period of time, based on the average number of events that occur in that period.
Here's how it works:
  • Consider a counting variable, such as the number of cars passing through a toll booth in an hour.
  • The Poisson PMF helps determine the chance of observing a specific number of events (e.g., cars) in the period.
  • The PMF is defined mathematically using the formula: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
In this formula: - \( P(X = k) \) represents the probability of observing \( k \) events.- \( \lambda \) is the average event rate.- \( e \) is approximately equal to 2.71828, the base of the natural logarithm.
Parameter Lambda
The parameter \( \lambda \) serves as a cornerstone for understanding the Poisson distribution. It signifies the expected number of occurrences within a specified interval.
Here's why \( \lambda \) is important:
  • \( \lambda \) holds a dual role; it's both the mean and the variance of the distribution. This means that not only does it tell how many events you expect, but it also says how much this expectation can vary.
  • Adjusting \( \lambda \) changes the distribution's behavior. A larger \( \lambda \) suggests a higher number of expected events, resulting in a wider distribution.
  • In practical instances, \( \lambda \) could refer to things like phone calls received at a call center per hour, or misprints in a book per page.
The beauty of \( \lambda \) is in its simplicity and versatility, making it a powerful parameter in evaluating probabilistic scenarios involving random, independent events.
Probability Computation
Calculating probabilities using the Poisson distribution requires a precise application of the Probability Mass Function. Once \( \lambda \) is defined, you can compute the probability of observing a specific number of events.
Here鈥檚 a simple walkthrough:
  • Identify the average rate \( \lambda \), which signifies the occurrences expected during the given interval.
  • Specify the number of actual events \( k \) you're interested in finding the probability for.
  • Use the PMF formula: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
    • - Here, \( e^{-\lambda} \) calculates the baseline probability for no events occurring.- \( \lambda^k \) then adjusts this baseline to accommodate \( k \) events.- Dividing by \( k! \) ensures the calculation accounts for the order of events.
Let's say a \( \lambda \) of 3 events per hour is expected, contemplating what is the chance of 2 events:
  • Input values into the formula:
  • \[ P(X = 2) = \frac{e^{-3} \times 3^2}{2!} \]
  • The calculated result offers the probability of exactly 2 events occurring in that hour.
This approach is essential when making predictions or assessing risks related to event occurrence rates.

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Most popular questions from this chapter

On the leeward side of the island of Oahu, in the small village of Nanakuli, about \(80 \%\) of the residents are of Hawaiian ancestry (Source: The Honolulu Advertiser). Let \(n=1,2,3, \ldots\) represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village of Nanakuli. (a) Write out a formula for the probability distribution of the random variable \(n\). (b) Compute the probabilities that \(n=1, n=2\), and \(n=3\). (c) Compute the probability that \(n \geq 4\). (d) In Waikiki, it is estimated that about \(4 \%\) of the residents are of Hawaiian ancestry. Repeat parts (a), (b), and (c) for Waikiki.

Which of the following are continuous variables, and which are discrete? (a) Speed of an airplane (b) Age of a college professor chosen at random (c) Number of books in the college bookstore (d) Weight of a football player chosen at random (e) Number of lightning strikes in Rocky Mountain National Park on a given day

For a binomial experiment, how many outcomes are possible for each trial? What are the possible outcomes?

The Honolulu Advertiser stated that in Honolulu there was an average of 661 burglaries per 100,000 households in a given year. In the Kohola Drive neighborhood there are 316 homes. Let \(r=\) number of these homes that will be burglarized in a year. (a) Explain why the Poisson approximation to the binomial would be a good choice for the random variable \(r .\) What is \(n\) ? What is \(p ?\) What is \(\lambda\) to the nearest tenth? (b) What is the probability that there will be no burglaries this year in the Kohola Drive neighborhood? (c) What is the probability that there will be no more than one burglary in the Kohola Drive neighborhood? (d) What is the probability that there will be two or more burglaries in the Kohola Drive neighborhood?

What does it mean to say that the trials of an experiment are independent?
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