Poisson Process Simulator
Poisson Process Simulator simulates a homogeneous Poisson process with rate \\( \lambda \\) over time interval \\([0, T]\\). Enter the rate parameter \\( \lambda \\) (events per unit time) and time horizon \\( T \\). The simulator generates event times using exponential inter-arrival times, visualized on a timeline with p5.js, with computational steps shown in MathJax.
Poisson Process Simulator
This simulator generates events for a homogeneous Poisson process with rate parameter \\( \lambda \\) (events per unit time) over a time interval \\([0, T]\\). Input \\( \lambda \\) (positive) and \\( T \\) (positive). The simulator uses exponential inter-arrival times (mean \\( 1/\lambda \\)) to generate event times, visualized as points on a timeline using p5.js. Computational steps, including inter-arrival times, are shown with MathJax. Share or embed results as needed.
Example 1: Low Rate
\\( \lambda = 1 \\), \\( T = 10 \\).
Result: Approximately 10 events with inter-arrival times \\( \sim \text{Exp}(1) \\).
Example 2: High Rate
\\( \lambda = 5 \\), \\( T = 5 \\).
Result: Approximately 25 events with inter-arrival times \\( \sim \text{Exp}(5) \\).
Example 3: Short Time Horizon
\\( \lambda = 2 \\), \\( T = 2 \\).
Result: Approximately 4 events with inter-arrival times \\( \sim \text{Exp}(2) \\).
What Is Poisson Process?
The Poisson process is a stochastic process where events occur randomly over time or space, and the number of events in a given interval follows a Poisson distribution. The key formula for the Poisson distribution, which describes the probability of observing k events in a given interval, is: P(X=k) = (e^(-λ) * λ^k) / k!. Here, λ represents the average rate of event occurrences, and e is the base of the natural logarithm (approximately 2.71828).Â