# Poisson/exponential process with step-wise decreasing rate

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• NotEuler
In summary, the conversation revolves around the concept of non-homogeneous Poisson distribution where the rate parameter decreases as a function of the number of events rather than time. The question posed is about the average number of events, variance, and other features of the distribution in this scenario. The conversation also touches upon the average time and variance until a certain number of events have taken place, which can be calculated by summing the mean and variance of each exponentially distributed time interval. The original question remains unanswered, seeking a simple way to calculate the mean number of events in a fixed time period.
NotEuler
So I've been stuck on trying to figure out the following problem. This is basically just for my own interest, not for a school problem or anything like that. It came out of trying to get my head around Poisson and exponential distributions.

Imagine you have a process where you have a fixed time T during which events happen. Now if the events all happened at a fixed rate λ, then the number of events that have taken place by time T would be Poisson distributed.

But what if the rate parameter decreases after each event? That is, we start with λ1, but after the first event, the rate parameter decreases to λ2<λ1. And after the second event, it decreases to λ3<λ2, and so on.

What, if anything can we now say about the average number of events that take place in the fixed time interval T? Or about the variance, or other features of the distribution of this number of events?

If I understand correctly, a non-homogeneous Poisson distribution would describe the process if the rate parameter decreased as a function of time. But in my example, it decreases as a function of the number of events. And the time intervals between the events are random.

A brief addition: It seems that it's quite easy to find the average time until n events have taken place.
Each time interval is exponentially distributed, with mean duration 1/λ1, 1/λ2 and so on.

Because of additivity of the mean, the mean until n events have taken place is then simply sum of the means: 1/λ1+1/λ2+...+1/λn.
Similarly, variance in the time until n events have taken place is 1/λ12+1/λ22+...+1/λn2.

Does this seem correct?

Still, my original intention was to figure out the mean number of events if the period of time is considered fixed. I'm curious if this can be done in a simple way.

## 1. What is a Poisson/exponential process with step-wise decreasing rate?

A Poisson/exponential process with step-wise decreasing rate is a stochastic process in which events occur randomly over time, following a Poisson distribution. The rate at which events occur decreases at specific points in time, creating a "step-wise" pattern.

## 2. How is this process different from a regular Poisson process?

In a regular Poisson process, the rate at which events occur remains constant over time. In a Poisson/exponential process with step-wise decreasing rate, the rate decreases at specific points in time, resulting in a non-constant rate.

## 3. What is the significance of using a step-wise decreasing rate in this process?

A step-wise decreasing rate allows for an event to occur more frequently at the beginning of the process and less frequently as time goes on. This can model real-life situations where there is an initial burst of activity followed by a decrease in frequency.

## 4. How is the rate of decrease determined in this process?

The rate of decrease is determined by the parameters of the process, such as the initial rate and the timing of the decreases. These parameters can be adjusted to fit the specific needs of the model being used.

## 5. What are some real-life applications of a Poisson/exponential process with step-wise decreasing rate?

This type of process can be used to model the spread of infectious diseases, the decay of radioactive materials, or the failure rate of mechanical systems. It can also be applied in finance to model stock prices or in economics to model consumer behavior.

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