Procedure

Sub-Experiment 1: Emergence of Poisson Process

This experiment demonstrates how a Poisson Process emerges from Repeated Bernoulli Trials by gradually increasing the number of trials ( n ). The Binomial distribution applies when we have a fixed number of events ( n ), each with a constant probability of success ( p ). But what if we don’t know the number of trials? Instead, we know the average rate of success per unit time, denoted as:

λ = n × p

Here, ( λ ) is the rate of successes per time unit. As ( n → ∞ ) and ( p → 0 ), while keeping ( λ ) constant, the Binomial distribution converges to a Poisson distribution.

Your task is to:

1. Start with a small value of ( n ) — the distribution should resemble a Bernoulli/Binomial behavior.
2. Gradually increase ( n ) while keeping ( λ ) fixed — observe how the two distributions on the chart begin to overlap.
3. Observe the decrease in the error metrics displayed below the chart as the value of ( n ) is increased, confirming the convergence.

Sub-Experiment 2: Merging of Poisson Processes

Next, we demonstrate the merging of Poisson processes using an analogy of radioactive emitters. We have 2 emitters with adjustable average rates.

• Let two independent emitters have average rates ( $λ_1$ ) and ( $λ_2$ ).
• Merging the two emitters results in a single Poisson process with a new rate: $\lambda = \lambda_1 + \lambda_2$.
• Reset the process reverts to the two independent streams with their original rates.

Your task is to:

1. Verify the merging phenomenon: Start the simulation and click "Merge". Observe in the "Observations" panel how the combined emission rate approaches the theoretical sum (λ₁ + λ₂).
2. Change the rates: Adjust the sliders for λ₁ and λ₂ and repeat the process to see that the principle holds for different values.

Sub-Experiment 3: Inter-arrival Times of the Poisson Process

Finally, we show that the inter-arrival times (i.e., the time between successive events) in a Poisson process follow an Exponential distribution. We also verify that these times are independent.

• If a Poisson process has rate ( λ ), the inter-arrival time ( T ) is an Exponential random variable with probability density: f(t) = λe⁻ˡᵗ.
• The independence of inter-arrival times means that the time until the next event does not depend on when the last event occurred.

Your task is to:

1. Set a rate (λ) using the slider, the inter arrival time you want to map and press Start.
2. Observe the Distribution Analysis chart: As events are recorded, watch the blue histogram of sampled inter-arrival times take the shape of the theoretical orange curve (the Exponential PDF).
3. Observe the Independence Analysis chart: Note that the scatter plot of consecutive inter-arrival times, (Tᵢ, Tᵢ₊₁), forms a random cloud. This indicates no correlation, and the best-fit line's slope should be near zero, confirming independence.