Exponential Distribution: For a Poisson process with an average event rate of \(\lambda\), the time between consecutive events (the inter-arrival time, \(T\)) is an Exponentially distributed random variable. Its PDF is given by \(f(t) = \lambda e^{-\lambda t}\).
Independence: A key property of the Poisson process is that its inter-arrival times are independent. The time you wait for the next event does not depend on how long you've already waited.
Testing Independence: We can visually test for independence by creating a scatter plot of consecutive inter-arrival times, \((T_i, T_{i+1})\). If the times are independent, the points should form a random "cloud" with no discernible pattern. The slope of a best-fit line through this cloud should be close to **zero**.
Procedure
Use the slider to set the Poisson rate (\(\lambda\)) and press **Start**.
Observe the timeline as events appear in real-time. The timeline shows absolute time and will pan as the simulation progresses.
The Distribution Analysisplot compares the histogram of waiting times to the theoretical Exponential PDF (orange line).
The Independence Analysis plot shows the scatter plot. Observe how the best-fit line remains nearly horizontal (slope ≈ 0).