This module demonstrates two key operations on Poisson processes: superposition (merging). When two independent Poisson streams are combined, the resulting stream is also a Poisson process.

Objective

• To visually and statistically verify that merging two independent Poisson processes with rates \(\lambda_1\) and \(\lambda_2\) results in a new Poisson process with a combined rate of \(\lambda_1 + \lambda_2\).
• To understand how a single Poisson process can be viewed as the superposition of multiple independent sub-processes.

Controls

• Rate \(\lambda_1\) (Emitter 1): Adjusts the average event rate for the first Poisson process.
• Rate \(\lambda_2\) (Emitter 2): Adjusts the average event rate for the second Poisson process.
• Merge / Reset Button: Toggles the simulation between two states: the merged state (one combined process) and the splited state (two independent processes).

Visualization

• Live Emission Animation: Provides a real-time visual of particles being emitted from the source(s), showing either two separate emitters or one merged emitter.
• Real-time Data Plots:
  • A line chart plots the number of events per second, with a vertical line indicating the moment of merging or splitting.
  • A histogram compares the distribution of observed event counts against the theoretical Poisson PMF for the current state.

Interpretation

• When in the merged state, the running average rate shown in the "Observations" panel should converge towards the theoretical sum \(\lambda_1 + \lambda_2\).
• When in the splitted state, the two processes run independently, and their individual behaviors are governed by their respective rates set by the sliders.
• The histogram of event counts should always closely match the shape of the theoretical Poisson distribution, whether in the merged or split state.