# Two independent Poisson processes (one discrete, one continuous)

• FireSail
In summary, the speaker is requesting help from the forum to find the probability mass function for a specific scenario involving two independent Poisson processes. They also mention their intuition to create a third process and calculate the probability from the joint distribution, but are unsure if it is the correct approach. The second part involves finding the conditional density and making predictions to minimize mean squared error. The speaker is unsure how to handle the combination of a discrete and continuous random variable. They suggest a possible solution and ask for guidance.

#### FireSail

Hi Guys,
I've used this forum as a great resource for a while now and it's always helped me out. Now I'm really stuck on something and was hoping you guys could help out. It's a pretty long question, but if you guys can just give me a general direction of what to do, I can go ahead and work it out for myself.

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Consider two independent Poisson processes N1 and N2 with rate parameters $$\lambda$$1 and $$\lambda$$2, respectively:

1. Find the prob. mass function for the number of events in N2 that occur before the first event after time 0 of N1 and identify what type of distribution it is.

So far my intuition to is to create a third process, N3 = N1 + N2, and then calculate the probability P(N3<0) from the joint distribution of N1 and N2. But I'm not sure this is the right way to do it.

The second part is trickier: Find the conditional density of the time of the first event after time 0 of N1 given that there are x events in N2 that occur before this first event of N1. Also, for a given x, how should you predict the time of the first event of N1 to minimize the mean squared error of your prediction?

The biggest problem I see is that I'm not sure how you're supposed to come up with a conditional probability if one random variable is discrete and the other is continuous. Can anyone point me in the right direction with this?

Thanks a ton you guys.

Might try this:
After time t, Probability for one event in process1 to occur over interval dt and N events in process 2 occurring after time t has elapsed should be the product:
λ1 exp(-λ1 t) exp(-λ2 t) (λ2 t)N/N! dt
Integrate over all positive t to get answer to question 1.

## 1. What is a Poisson process?

A Poisson process is a type of stochastic (random) process that models the occurrence of events over time. It is characterized by the assumptions that the events occur independently of each other and that the rate of occurrence is constant. It is often used to model real-world phenomena such as arrival times, failure rates, and radioactive decay.

## 2. What is the difference between a discrete and a continuous Poisson process?

A discrete Poisson process is one where the events occur at distinct, discrete points in time. This means that there is a fixed time interval between events. A continuous Poisson process, on the other hand, has events that occur at any point in time, and the time between events can vary.

## 3. How do you calculate the probability of a certain number of events occurring in a given time interval for a Poisson process?

The probability of a certain number of events occurring in a given time interval for a Poisson process can be calculated using the Poisson distribution formula: P(X = x) = (λ^x * e^-λ) / x!, where λ is the average rate of occurrence and x is the number of events.

## 4. Can there be correlations between events in a Poisson process?

No, the assumption of independence between events is a fundamental characteristic of a Poisson process. This means that the occurrence of one event does not affect the likelihood of another event happening.

## 5. How is a Poisson process used in scientific research?

Poisson processes are commonly used in scientific research to model and analyze data in various fields such as biology, physics, and economics. They can be used to predict the likelihood of events occurring, assess the effectiveness of interventions, and make inferences about underlying processes. They are also useful in simulation studies and statistical analyses.