# Proof of Exponential Interarrival times times

• mmehdi
In summary, the conversation discusses the proof that Poisson processes have exponential interarrival time, and that exponential interarrival times will always be a Poisson process. The fundamental assumption of the Poisson distribution is that the probability of a single arrival is constant and the probability of multiple arrivals is negligible. The proof of this hypothesis can be found in probability texts or can be treated as a differential equations problem with an exponential function as the solution. The conversation also explores the relationship between exponential and Poisson distributions, with the suggestion that the integration of the gamma distribution yields a Poisson distribution. However, the integration process is complex.
mmehdi
The proof that poisson process has exponential interarrival time is common place. The proof which i am trying to do is that, exponential interarrival times will always be poisson process, its like the reverse of the earlier proof. Could you help me with that.

The fundamental assumption of the Poisson distribution is that, for some very small time interval, the probability of a single arrival is a constant and the probability of more than one arrival in that time interval is so small it can be ignored. The proof that that hypothesis leads to the Poisson distribution is given in any good probability text that discusses the Poison distribution. You can also treat that as a differential equations problem (the rate of change of total arrivals is constant) that has an exponential function as solution.

I am not sure if I truly follow you. The proof that you can only have one arrival in one interval and the probability of getting two arrival is zero, is done through taylor series expansion. But how does it show that the exponenttial interarrival shall always satisfy the poisson properties.

What I tried is the total time, or waiting time let's say for the second arrival is Gamma Distribution. So if we integrate the gamma distribution it shall give us a poisson distribution. However the integration is winding, but it does yield the poisson distribution at the end.

## 1. What is Proof of Exponential Interarrival times?

Proof of Exponential Interarrival times is a mathematical concept that is used to model the arrival of events or occurrences that happen randomly and independently of each other. It is often applied in the study of queueing systems, where the time between arrivals of customers or events follows an exponential distribution.

## 2. How is Proof of Exponential Interarrival times useful in science?

Proof of Exponential Interarrival times is useful in science because it can be used to model and analyze real-world phenomena that involve the arrival of random events. This can include the study of natural phenomena such as earthquakes, as well as human-made systems like transportation and telecommunication networks.

## 3. What are the key assumptions in Proof of Exponential Interarrival times?

The key assumptions in Proof of Exponential Interarrival times are that the events or occurrences happen randomly and independently of each other, and the time between arrivals follows an exponential distribution. Additionally, it is assumed that there is no correlation between the time of arrival of one event and the next, and the probability of an event occurring is the same for all time intervals.

## 4. How is Proof of Exponential Interarrival times related to the Poisson process?

Proof of Exponential Interarrival times is closely related to the Poisson process, which is a mathematical model for the arrival of events in a given time interval. The Poisson process assumes that the number of events that occur in a time interval follows a Poisson distribution, and the time between arrivals follows an exponential distribution.

## 5. What are some real-world applications of Proof of Exponential Interarrival times?

Proof of Exponential Interarrival times has many real-world applications, including the study of natural phenomena such as the occurrence of earthquakes and the arrival of meteorites. It is also used in the study of human-made systems like transportation networks, communication systems, and computer networks. Additionally, it is used in market analysis, financial modeling, and risk management.

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