This experiment demonstrates two fundamental properties of a Poisson process by running a large number of simulated trials.

Objective

• To show that the inter-arrival time \(T_i\) in a Poisson process follows an Exponential distribution with PDF \(f(t) = \lambda e^{-\lambda t}\).
• To verify that consecutive inter-arrival times, \(T_i\) and \(T_{i+1}\), are statistically independent.

Understanding the Notation

Imagine events occurring on a timeline. We use the following notation to describe them:

• Event Times (t_i): These are the absolute moments when events occur. For example, t₁ is the time of the 1st event, and t₂ is the time of the 2nd event.
• Inter-arrival Time (T_i): This is the waiting time, or the duration between consecutive events. It is defined as T_i = t_i - t_i-1. For example, T₂ is the time that passes between the 1st and 2nd events (i.e., t₂ - t₁).

This experiment analyzes the properties of these inter-arrival times (T_i).

Procedure

1. Set the Poisson Rate (\(\lambda\)), which is the average number of events per second.
2. Choose which inter-arrival time to analyze using the dropdown (e.g., the 1st, 2nd, or 3rd). This selects the index i for T_i.
3. Select the Number of Trials to run. A higher number will produce a more accurate result.
4. Click "Run Simulation". The simulation will run all trials instantly.
5. Analyze the Distribution: The left chart shows a histogram of the collected T_i values. Observe how closely the blue histogram matches the theoretical orange Exponential curve.
6. Analyze Independence: The right chart plots pairs of consecutive inter-arrival times, (T_i, T_i+1). If the points form a random, shapeless cloud with a best-fit line slope near zero, the times are independent.