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Chapter 6: Problem 1

Flies and wasps land on your dinner plate in the manner of independent Poisson processes with respective intensities \(\lambda\) and \(\mu\). Show that the arrivals of flying objects form a Poisson process with intensity \(\lambda+\mu\).

Short Answer

Expert verified
The arrivals of flying objects form a Poisson process with intensity \(\lambda + \mu\).

Step by step solution

01

Understand the Poisson Process

A Poisson process is a stochastic process where events happen continuously and independently at a constant average rate. Here, flies arrive at an average rate of \(\lambda\), and wasps arrive at an average rate of \(\mu\).
02

Consider the Combined Process

Since the events (flies and wasps landing) are independent, we need to find the rate (intensity) of both arriving together. Let \(N(t)\) be the total number of flying objects (both flies and wasps) arriving by time \(t\).
03

Use the Additive Property

The number of arrivals in a Poisson process with rates \(\lambda\) and \(\mu\) are independent Poisson random variables. The sum of two independent Poisson random variables with rates \(\lambda\) and \(\mu\) is another Poisson random variable with rate \(\lambda + \mu\).
04

Write the Mathematical Expression

By the additive property, the total rate \(\lambda_{total}\) for the combined process is given by:\[ N(t) \sim \text{Poisson}((\lambda + \mu)t) \]This shows that \(N(t)\) is a Poisson process with intensity \(\lambda + \mu\).
05

Conclusion

From steps 1 to 4, we have shown that the sum of two independent Poisson processes remains a Poisson process, now with a combined intensity of \(\lambda + \mu\).

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Key Concepts

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

Independent Processes
Independent processes are all about events happening without affecting each other. In the context of our exercise, flies and wasps landing on a plate are independent processes. This means the arrival of a fly doesn't change the chances of a wasp landing.
This independence is crucial because it allows us to treat each arrival separately when modeling them mathematically.
  • Each type of insect arrives according to its own schedule.
  • No interaction or dependency exists between their arrivals.
Understanding independence becomes vital in combining these processes, as we'll see when we talk about Poisson processes.
Stochastic Process
A stochastic process describes a system that evolves over time in a way that is not deterministic, meaning outcomes arise with probability rather than certainty.
In our situation, the arrival of flies and wasps is modeled by a stochastic process.
  • The arrivals are random and follow a probability distribution.
  • This randomness is captured by modeling them as Poisson processes.
In essence, each insect's arrival is a small event contributing to the overall stochastic nature of the system.
We use stochastic processes to model situations where outcomes can vary even if the overall average behavior is known.
Rate (Intensity)
The rate or intensity in a Poisson process represents the average number of times an event is expected to occur over a period of time.
In our exercise,

  • \(\lambda\)
  • mu
mu
When we talk about combining the Poisson processes of flies and wasps, we add these rates together to find the total rate.
  • This principle is due to the additive property of Poisson processes, which states that two independent Poisson processes with rates \(\lambda\) and \(\mu\) can be combined into a single Poisson process with rate \(\lambda + \mu\).
This shows how the combined activity over time remains predictable on average, even though it is random in nature.
Random Variables
Random variables are variables whose possible values are numerical outcomes of a stochastic process.
In the Poisson process situation of our exercise, we deal with random variables when considering the number of flies or wasps landing on the plate.
  • Each individual landing is a random event modeled by a Poisson random variable.
  • This variable predicts how many landings occur within a specific time.
The key characteristic of these variables is that they follow a probability distribution, which in our case, is a Poisson distribution.
The use of random variables allows us to apply statistical and mathematical methods to predict and analyze outcomes, explaining the randomness involved in arrival processes.

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Most popular questions from this chapter

Let \(X\) and \(Y\) be Markov chains on the set \(Z\) of integers. Is the sequence \(Z_{n}=X_{n}+Y_{n}\) necessarily a Markov chain?

A village of \(N+1\) people suffers an epidemic. Let \(X(t)\) be the number of ill people at time \(t\). and suppose that \(X(0)=1\) and \(X\) is a birth process with rates \(\lambda_{i}=\lambda i(N+1-i)\). Let \(T\) be the length of time required until every member of the population has succumbed to the illness. Show that $$ \mathrm{E}(T)=\frac{1}{\lambda} \sum_{k=1}^{N} \frac{1}{k(N+1-k)} $$ and deduce that $$ \mathbb{E}(T)=\frac{2(\log N+\gamma)}{\lambda(N+1)}+\mathrm{O}\left(N^{-2}\right) $$ where \(\gamma\) is Euler's constant. It is striking that \(\mathbb{E}(T)\) decreases with \(N\), for large \(N\).

Another diffusion model. \(N\) black balls and \(N\) white balls are placed in two ums so that each contains \(N\) balls. After each unit of time one ball is selected at random from each urn, and the two balls thus selected are interchanged. Let the number of black balls in the finst um denote the state of the system. Write down the transition matrix of this Markov chain and find the unique stationary distribution. Is the chain reversible in equilibrium?

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The proof copy of a book is read by an infinite sequence of editors checking for mistakes. Each mistake is detected with probability \(p\) at each reading: between readings the printer corrects the detected mistakes but introduces a random number of new errors (errons may be introduced even if no mistakes were detected). Assuming as much independence as usual, and that the numbers of new errors after different readings are identically distributed, find an expression for the probability generating function of the stationary distribution of the number \(X_{n}\) of errors after the \(n\)th editor-printer cycle, whenever this exists. Find it explicitly when the printer introduces a Poisson-distributed number of errors at each stage.
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