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Chapter 12: Problem 1

In each of the following examples, try to indicate whether the Poisson process would be a good model. a. The times of bankruptcy of enterprises in the United States. b. The times a chicken lays its eggs. c. The times of airplane crashes in a worldwide registration. d. The locations of worngly spelled words in a book. e. The times of traffic accidents at a crossroad.

Short Answer

Expert verified
Only example e is a good fit for a Poisson process.

Step by step solution

01

Understanding Poisson Process

The Poisson process is typically used to model random events that happen independently and at a constant average rate. It's suitable when we want to count how many events happen over a fixed period.
02

Evaluating Example a

For 'The times of bankruptcy of enterprises in the United States', a Poisson process might not be ideal since bankruptcies can be influenced by broader economic factors, making them not independent events.
03

Evaluating Example b

For 'The times a chicken lays its eggs', a Poisson process may not be appropriate. Egg-laying is biologically regulated and not independent or random at a constant rate over time.
04

Evaluating Example c

For 'The times of airplane crashes in a worldwide registration', this might not be a good fit for a Poisson process as crashes are not truly independent events and are influenced by numerous factors.
05

Evaluating Example d

For 'The locations of wrongly spelled words in a book', a Poisson process might not be ideal. Errors are likely to be more common in sections written quickly or not edited thoroughly, thus not random or independent.
06

Evaluating Example e

For 'The times of traffic accidents at a crossroad', a Poisson process can be suitable if accidents are mostly independent events occurring at a constant average rate.

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Key Concepts

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

Random Events
Random events are occurrences that happen without a predictable pattern. In our everyday life, many phenomena can appear random because they can't be anticipated exactly, such as the roll of a dice or the weather on a specific day. In the context of a Poisson process, events are considered random when they can't be predicted in detail beforehand.

This randomness is crucial when modeling events using the Poisson process because:
  • It helps in approximating the real-world unpredictability in the timing and number of occurrences.
  • Each event is thought to occur independently of others, aligning with the unpredictable nature of random happenings.
Ensuring randomness in a Poisson process model ensures that we reflect natural uncertainties accurately, providing insights into behaviors that might otherwise seem chaotic.
Independence
Independence in statistical terms means that the occurrence of one event does not affect the likelihood of other events occurring. This concept is central to the Poisson process.

For example:
  • The chance of a chicken laying an egg today is not increased by the fact that it laid an egg yesterday, assuming the Poisson model governs the process.
  • Accidents at a crossroad, in an ideal model, occur independently from each other; one accident does not alter the statistical likelihood of another happening.
If events are not independent, then the Poisson process may not be the best model to use. Reflecting on whether real-world scenarios show dependence or independence helps in deciding the suitability of using the Poisson process.
Constant Rate
A constant rate implies that events occur on average at a steady pace over time. This element is a pillar of the Poisson process model and assumes that the rate at which events happen does not vary, even if the absolute timing cannot be predicted.

Consider:
  • A crossroad experiencing traffic accidents at roughly the same rate every year, with no seasonal variations or influencing factors.
  • The concept of constancy supports the model where the number of events is consistently described over equal intervals (e.g., accidents per month).
When examining if a scenario fits a Poisson process, it's crucial to verify that the rate is genuinely stable over time. Deviations from constancy might prompt re-evaluation of the model being used.
Event Modeling
Event modeling is the process of using mathematical models to predict or simulate how often and when events occur. The Poisson process is a popular method of event modeling, especially when dealing with rare events across a period.

Why use event modeling?
  • It helps in setting expectations, such as forecasting how many book errors might arise in different sections.
  • The models can guide decision-making, like determining needed safety measures at a busy intersection.
The choice of a Poisson process for event modeling depends on accurately meeting its criteria - random occurrence, independence, and constant rate. Mastery of these concepts helps apply the Poisson model effectively, ensuring that modeled outcomes are truly reflective of the real world.

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Most popular questions from this chapter

Let \(X\) be a random variable with a Pois \((\mu)\) distribution. Show the following. If \(\mu<1\), then the probabilities \(\mathrm{P}(X=k)\) are strictly decreasing in \(k\). If \(\mu>1\), then the probabilities \(\mathrm{P}(X=k)\) are first increasing, then decreasing (cf. Figure 12.1). What happens if \(\mu=1\) ?

The number of customers that visit a bank on a day is modeled by a Poisson distribution. It is known that the probability of no customers at all is \(0.00001\). What is the expected number of customers?

A certain brand of copper wire has flaws about every 40 centimeters. Model the locations of the flaws as a Poisson process. What is the probability of two flaws in 1 meter of wire?

Consider the one-dimensional Poisson process with intensity \(\lambda\). Show that the number of points in \([0, t]\), given that the number of points in \([0,2 t]\) is equal to \(n\), has a \(\operatorname{Bin}\left(n, \frac{1}{2}\right)\) distribution. Hint: write the event \(\\{N([0, s])=k, N([0,2 s])=n\\}\) as the intersection of the (independent!) events \(\\{N([0, s])=k\\}\) and \(\\{N((s, 2 s])=n-k\\}\).

Let \(X\) be a Poisson random variable with parameter \(\mu\). a. Compute \(\mathrm{E}[X(X-1)]\). b. Compute \(\operatorname{Var}(X)\), using that \(\operatorname{Var}(X)=\mathrm{E}[X(X-1)]+\mathrm{E}[X]-(\mathrm{E}[X])^{2}\).
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