Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Poisson Process: Properties

Description

This lecture covers the properties of Poisson processes, including the inter-arrival time, counting process, and stochastic models for communications. It discusses the probability of having m arrivals in a given time interval, the exponential distribution, and the Erlang distribution.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (15)

Poisson point process

In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field.

Exponential distribution

In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

Compound Poisson process

A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate and jump size distribution G, is a process given by where, is the counting variable of a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of When are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process.

Erlang distribution

The Erlang distribution is a two-parameter family of continuous probability distributions with support . The two parameters are: a positive integer the "shape", and a positive real number the "rate". The "scale", the reciprocal of the rate, is sometimes used instead. The Erlang distribution is the distribution of a sum of independent exponential variables with mean each. Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of .

Gamma distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: With a shape parameter and a scale parameter . With a shape parameter and an inverse scale parameter , called a rate parameter. In each of these forms, both parameters are positive real numbers.

Related lectures (35)

Continuous Time Markov Chains

Introduces continuous time Markov chains on a finite state space with exponential waiting times and jump probabilities.

Poisson processes

Covers the properties and construction of Poisson processes from i.i.d. Exp(X) random variables, emphasizing the importance of the process rate and jump time distributions.

Point Processes: Spatial Analysis

Explores point processes in spatial analysis, focusing on spatial object dissemination and pattern detection.

Stochastic Models for Communications Log in to Mediaspace to watch this video

Explores stochastic models for communications, covering mean, variance, characteristic functions, inequalities, various discrete and continuous random variables, and properties of different distributions.

Rainfall Models: Deterministic vs Stochastic Log in to Mediaspace to watch this video