Poisson process, question about the definition.

In summary, there is a discussion about proving point (2) from point (3) in the definition of the Poisson process. The concern is that there seems to be something hidden in the Poisson distribution that allows for the properties to hold. It is noted that for the Poisson distribution, the probability of 0 events can be written as the product of the probabilities of 0 events in each interval, leading to independence. However, this is not the case for the geometric distribution. It is suggested that this property of functions may have a name.
  • #1
bobby2k
127
2
Hi, I have a question about the definition of the poisson process. Check out the definition here:

poisson.png
Would you say that one can prove point (2) from point (3)?

The reason I have some discomfort about this is that something seems to be hidden in the poisson distribution to make it all work?

For instance, from point (3) I am able to prove independent increments if we are looking at the probability of 0 events.

For instance, let's say you have two disjoint intervals [itex]\Delta t[/itex] and [itex]\Delta s[/itex].
Then from point (3) we get that probability of 0 events in the union of these two intervals is [itex]e^{-\lambda(\Delta t + \Delta s)}[/itex], but this can be written as [itex]e^{-\lambda\Delta t}*e^{-\lambda \Delta s}[/itex]. So since we are able to multiplicate each marginal probability, in this case we got independence directly from 3.

What is it I am not seeing?

EDIT: Also look at this fact for 1 event in the interval.
From (3) the probability of 1 event in the interval [itex]\Delta t + \Delta s[/itex] is: [itex]e^{-\lambda (\Delta t + \Delta s)}*(\lambda(\Delta t + \Delta s))[/itex].

But if we assume independce and (3) we get that this probability can also be calculated as the probability for 1 in the first interval and 0 in the last, plus the probability of 0 in the first interval and 1 in the last:

[itex]e^{-\lambda \Delta t}*e^{-\lambda \Delta s}*(\lambda \Delta s)+e^{-\lambda \Delta s}*e^{-\lambda \Delta t}*(\lambda \Delta t)=e^{-\lambda(\Delta t + \Delta s)}*(\lambda(\Delta t + \Delta s))[/itex].
We get the same. And hence it does look like there is something in the poisson distribution that makes this all works? I mean, if we kept point (1) and (2) and changed poisson with geometric(the probability distribution would now be independent of the langth of the interval aswell), we would not get these properties. So it seems like they couldn't just pick a distribution and put it in the definition. Can it be that a distribution would have to have some multiplicative property in regards to time intervals?
 
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  • #2
Or would you say that when we have chosen condition (2) we have chosen to give an implicit condition for the probability distribution? This condition states that:
[itex]P(N(\Delta s + \Delta t)=n) = \Sigma_{i,j|i+j=n} [P(N(\Delta s)=i)*P(N(\Delta t)=j)] [/itex].
And the poisson distribution with parameter [itex]\lambda \Delta T[/itex], where [itex]\Delta T[/itex] is the time interval, have this property, so it is ok to use it. But the geometric distribution does not have this property, hence we can not use it?

If this is correct, does this property of functions have a name?
 
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What is a Poisson process?

A Poisson process is a mathematical model used to describe the occurrence of events over a certain time period. It assumes that events occur randomly and independently of each other, with a constant average rate of occurrence. It is named after the French mathematician Siméon Denis Poisson.

How is a Poisson process defined?

A Poisson process is defined by three key characteristics: 1) the events occur randomly and independently of each other, 2) the average rate of occurrence is constant, and 3) the time between events follows an exponential distribution. It can also be described using a counting process, where the number of events that occur in a given time interval follows a Poisson distribution.

What are some real-world examples of a Poisson process?

A Poisson process can be used to model a variety of natural and man-made phenomena, such as the number of customers arriving at a store, the number of earthquakes in a certain region, or the number of phone calls received by a call center. It can also be used in traffic analysis, queuing theory, and inventory management.

What are the limitations of a Poisson process?

A Poisson process assumes that events occur independently and at a constant rate, which may not always be the case in real-world situations. It also does not account for the impact of external factors on the occurrence of events. Additionally, it is not suitable for modeling events that have a significant impact on each other, such as disease outbreaks or natural disasters.

How is a Poisson process related to other probability distributions?

A Poisson process is closely related to the Poisson distribution, which is a discrete probability distribution that describes the probability of a certain number of events occurring in a fixed time interval. It is also related to the exponential distribution, which describes the time between events in a Poisson process. Furthermore, the Poisson process is a special case of a more general class of stochastic processes known as Markov processes.

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