13. Bernoulli Process

What this lecture covers

This lecture begins the last major topic of MIT 6.041: random processes, which model random phenomena evolving over time rather than a single random variable in isolation. It introduces the simplest such process, the Bernoulli process, a sequence of independent, identically biased coin flips, and asks two natural questions: given a fixed amount of time, how many arrivals occur, and given a fixed number of arrivals, how long does it take?

The lecture sits at the start of the random processes unit and lays groundwork for the continuous-time analogue (the Poisson process) covered next, as well as for Markov chains later in the course. After watching, you should be able to state the Bernoulli process's assumptions, compute distributions for arrival counts and waiting times, explain the memorylessness property and when it does or doesn't apply, and describe what happens when Bernoulli processes are split or merged.

Key ideas

• Bernoulli process definition: an infinite sequence of independent trials, each with the same success probability p, such as repeated coin flips or a simplified model of arrivals to a facility.
• Number of arrivals in n slots is binomial: with mean n*p and variance n*p*(1-p).
• Time until the first arrival is geometric: with parameter p, mean 1/p, counting trials up to and including the first success.
• Memorylessness: given only what happened up to a point determined without foresight, the future is still a fresh sequence of independent Bernoulli trials, regardless of what came before, including runs of successes or failures.
• Sum of independent geometrics: the time until the k-th arrival, Yk, is the sum of k independent geometric inter-arrival times, with E[Yk] = k/p and Var(Yk) = k * Var(T); its exact PMF is derived from independence between the count of arrivals before time t and the event of an arrival at time t.
• Splitting a Bernoulli process: routing each arrival independently with probability q to one of two destinations produces two independent Bernoulli processes with parameters p*q and p*(1-q).
• Merging two independent Bernoulli processes: recording an arrival whenever either process has one produces another Bernoulli process, with success probability p + q - p*q.
• Infinite sequences have zero individual probability: any specific infinite sequence of coin flip outcomes, including all-heads, has probability 0, analogous to any single point having zero probability under a continuous distribution.

Walkthrough

Defining the Bernoulli process (2:03)

The lecture defines the Bernoulli process as a sequence of independent trials with constant success probability p, giving examples (lottery tickets, financial market up/down days, arrivals to a bank or server), and notes the two key assumptions: independence across trials and a constant p, which may only hold approximately over limited time windows in real applications.

Two views of a random process (6:08)

The lecture presents the Bernoulli process first as a collection of random variables with known individual and joint distributions (simplified here because independence makes joint PMFs a simple product), and second as a single experiment whose sample space is the set of all infinite 0/1 sequences, showing that any specific infinite sequence, including all successes, has probability zero.

Binomial counts and geometric waiting times (17:22)

For a fixed number of time slots, the number of arrivals is binomial with mean np and variance np(1-p). For the time until the first arrival, the lecture derives the geometric distribution, with mean 1/p.

The memorylessness property (21:30)

Using the idea of being "called in to watch" at a time determined without foresight, the lecture argues that whatever happened before that point (however unusual) does not affect the statistics of what happens afterward, which remains a sequence of independent Bernoulli trials. It contrasts this with a subtle example (the length of the first string of losing days), showing why a naive geometric-minus-one guess is wrong and deriving the correct geometric answer by choosing the right point to start watching.

Time until the k-th arrival (33:46)

The lecture defines inter-arrival times T1, T2, T3, ... as independent geometric random variables and shows that the time until the k-th arrival, Yk, is their sum. It derives the exact PMF of Yk directly (rather than via repeated convolution) using independence between the number of arrivals in the first t-1 slots and the event of an arrival at slot t, and gives its mean and variance as k times those of a single geometric.

Splitting and merging Bernoulli processes (43:03)

The lecture shows that routing each arrival to one of two destinations via an independent coin flip splits a Bernoulli process into two independent Bernoulli processes with parameters p*q and p*(1-q). It then shows the reverse: merging two independent Bernoulli processes by recording an arrival whenever either has one produces another Bernoulli process with parameter p + q - p*q, previewing ideas that return in continuous time.

Before you watch

• Be comfortable with the binomial and geometric distributions and their mean and variance formulas.
• Review the concept of independence between random variables, since the Bernoulli process's key assumption is independence across time slots.

Check your understanding

1. What two assumptions define a Bernoulli process, and when might the constant-p assumption fail in a real system?
2. Explain the memorylessness property in your own words, and why it fails if the observer has foresight.
3. Why is the length of the first string of losing days geometric, rather than "geometric minus one"?
4. Derive the PMF of the time until the k-th arrival using independence between two events.
5. Why does splitting a Bernoulli process by an independent routing coin produce two Bernoulli processes rather than some other kind of process?