# How Do You Determine Which Independent Poisson Process Occurs First?

• circa415
In summary, the conversation discusses how to calculate the probability of one poisson process arriving before another, and various methods for doing so. These include defining a new random variable and using the joint distribution, as well as working with the time of the nth arrival and using the memorylessness property. The result for n=1 is also mentioned.
circa415
I'm not sure how to get started with this:

Suppose you have two independent poisson processes, X ~ Po(Lambda1) Y ~Po (Lambda2). How would you figure out the probability of one coming before the other?

Thanks

circa415 said:
How would you figure out the probability of one coming before the other?
What are you modeling, arrival times or number of arrivals? Poisson is generally used for the latter.

In problems involving a comparison between two random variables (e.g. X < Y) one usually needs to define a new random variable, say Z = X - Y, then calculate Pr{Z < 0} from the joint distribution of X and Y.

thanks for the tip

I'm modeling arrival times for a poisson process. Ex: the time it takes for an insurance company to receive n claims. For instance, what is the probability that One insurance company receives n claims before another insurance company receives n claims

Last edited:
circa415 said:
For instance, what is the probability that One insurance company receives n claims before another insurance company receives n claims
AFAIK the Poisson distribution describes the number of arrivals in a fixed time frame. This makes me think that you may want to reformulate your question as the number of arrivals in the first process being less (or greater) than those in the second process within a given time period.

A Po(Lambda) process models a process whose jumps (arrivals) occur at rate Lambda - probability of a jump in time dt is Lambda * dt to highest order in dt (http://en.wikipedia.org/wiki/Poisson_process" .) You could use the fact that the distance between these jumps are independent and exponentially distributed. So the time to the n'th jump (n'th claim) is the sum of n independent exponential r.v.s, which is a chi(2n)-squared distribution.

You could also work out the result for n = 1, and then extend to larger n. If t(n) is the time of the n'th arrival of X & Y combined, then look at the process Z(n) = X(t(n))-Y(t(n)). You can use the memorylessness property to show that this will be a random walk, and is given by the binomial distribution for each n. To say that the first n arrivals of X occur before the first n arrivals of Y is equivalent to Z(2n-1)>=0, which you can then get from the binomial distribution.

Do you know the result for n=1?

Last edited by a moderator:

## 1. What is an Independent Poisson Process?

An Independent Poisson Process is a type of stochastic process where events occur randomly and independently of each other. It is characterized by a constant rate of occurrence and the probability of an event happening within a given time interval is equal to the length of the interval multiplied by the rate.

## 2. How does an Independent Poisson Process differ from a regular Poisson Process?

The main difference between an Independent Poisson Process and a regular Poisson Process is that in an independent process, the occurrence of one event does not affect the probability of another event happening. In a regular Poisson Process, the occurrence of one event decreases the probability of another event happening within the same time interval.

## 3. What are some real-world examples of Independent Poisson Processes?

Some real-world examples of Independent Poisson Processes include the arrival of customers at a store, the number of emails received in a day, and the number of phone calls received by a call center. These events occur randomly and independently of each other, making them good examples of Independent Poisson Processes.

## 4. How is the rate of an Independent Poisson Process determined?

The rate of an Independent Poisson Process is determined by the expected number of events that occur within a specific time interval. This rate remains constant throughout the process and is used to calculate the probability of an event occurring within a given time interval.

## 5. What are some applications of Independent Poisson Processes in science?

Independent Poisson Processes have many applications in science, including modeling the spread of diseases, analyzing radioactive decay, and predicting the number of accidents in a certain area. They are also commonly used in queueing theory to analyze waiting times and service rates in various systems.

• Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
8
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
12
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
28
Views
3K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
4K
• Set Theory, Logic, Probability, Statistics
Replies
37
Views
4K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K