Procedure

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 distribution transforms into a Poisson distribution.
3. Observe the decrease in the error of the two distributions as the value of n is increased.

Merging and Splitting of Poisson Processes

Next, we demonstrate the superposition (merging) and splitting of Poisson processes using an analogy of radioactive emitters. We have 2 emitters with average rates which can be adjusted.

• Let two independent emitters have average rates ( λ_1 ) and ( λ_2 ).
• Merging the two emitters results in a single Poisson process with rate: λ = λ_1 + λ_2
• Splitting the process probabilistically assigns each event to either of the original sources, effectively recreating the processes with rates ( λ_1 ) and ( λ_2 ).

Your task is to:

1. Verify the phenomenon - by noting the new average rates of emssion before and after splitting/merging.
2. Check for by changing value of λ_1 and λ_2 - change the value of λ_1 and λ_2 and verify the phenomenon again.
3. Check the splitted distribution - check the new distributions show poisson process with the correct average rates.

Interarrival Times of the Poisson Process

Finally, we show that the interarrival times (i.e., the time between successive events) in a Poisson process follow an Exponential distribution. Specifically:

• If the Poisson process has rate ( λ ),
• Then the interarrival time ( T ) is an Exponential random variable with probability density: f_T(t) = λ × exp(–λt),   t ≥ 0

We simulate this behavior and verify that the histogram of interarrival times matches the exponential distribution.