Homework 8

Analogies between the Bernoulli Process Simulation and the Random Walk Model

Introduction

Both the Bernoulli process simulation and the random walk model describe systems that evolve through a sequence of independent binary outcomes. In the Bernoulli process, each trial can result in “success” or “failure,” while in the random walk, each step can go “up” or “down.” Despite the different interpretations, both models are mathematically equivalent, as they are built upon the same probabilistic foundation, the Bernoulli trials.

This section highlights the analogies, differences, and mathematical relationships between these two models, showing how concepts such as binomial coefficients, Pascal’s triangle, binomial expansion, and even Fibonacci sequences naturally arise from their shared structure.

Shared foundations both the homeworks 6 and 7 share: The Bernoulli Process

In both homeworks, the process is governed by independent trials with two possible outcomes:

• success (or secure week) with probability $q=1-p$,

• failure (or breach) with probability p.

Each trial Xi​ is a Bernoulli random variable, taking value 1 for success and 0 for failure.

The total number of successes in $n$ trials follows:

In the Bernoulli simulation, the Law of Large Numbers (LLN) was illustrated by showing that the sample mean

converges to the true probability q as $n \to \infty$. In the random walk model, the same trials determine step direction, mapping directly to the binary outcome structure of the Bernoulli process.

Analogies

We can consider the Random Walk as a Centered Bernoulli Process. In fact, the random walk version simply redefines the outcomes: instead of Xi ∈ {0,1} it uses Yi ∈ {-1,+1}, with:

Yi = 2Xi - 1

Therefore, we'll have:

Basically, the distribution of Sn is a re-scaled binomial distribution: both models generate the same combinatorial structure, changing only the interpretation. Also, both models exhibit the same convergence patterns as n grows: empirical frequencies stabilize around theoretical probabilities, illustrating the Law of Large Numbers.

Mathematical structures and needed combinatorial links

The probability of obtaining exactly k successes is given by the binomial formula:

Each term $\binom{n}{k}$ counts the number of ways to arrange $k$ successes among $n$ independent trials. The same coefficient appears in the random walk distribution through the mapping Sn​=2k−n.

Also, Binomial coefficients can be represented recursively in Pascal’s Triangle (also known as Tartaglia's Triangle):

 

Each row of the triangle corresponds to the expansion of:

In our probabilistic context, if $a = q$ and $b = p$, the expansion describes the total probability distribution over all possible outcomes of $n$ Bernoulli trials.

Moreover, the connection with the Fibonacci sequence emerges from the combinatorial relationships among binomial coefficients. One example can be:

This shows how Fibonacci numbers can be expressed as sums of specific binomial coefficients, linking probabilistic combinatorics with classical number sequences. While not directly simulated in our homeworks, this illustrates that the same combinatorial structures underpin a wide variety of discrete stochastic and recursive systems.

Differences between the two homeworks

So, although both rely on Bernoulli processes, their goals differ:

• The Bernoulli simulation (previous homework) aimed to illustrate convergence of the sample mean to its theoretical expectation (Law of Large Numbers).

• The random walk simulation studied the distribution of cumulative scores, showing convergence of empirical frequencies to a binomial probability mass function.

In short:

• HW on Bernoulli (6th) → asymptotic convergence (LLN)

• HW on Random Walk (7th) → distributional equivalence (Binomial/Combinatorial structure)

Conclusions

Both the Bernoulli and random walk simulations share the same probabilistic DNA, differing only by interpretation and focus. Their behavior is governed by the binomial model, which connects directly to fundamental combinatorial constructs such as Pascal’s triangle and binomial expansion. These relationships highlight how simple probabilistic mechanisms generate deep mathematical patterns, unifying discrete probability, combinatorics, and number theory within a single conceptual framework.

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