Bernoulli process

Definition

Let $p \in [0, 1]$. A Bernoulli process with parameter $p$ is a stochastic process $(X_n)_{n \ge 1}$ of i.i.d. random variables with the Bernoulli distribution $\operatorname{Bernoulli}(p)$: $$P(X_n = 1) = p \quad \text{and} \quad P(X_n = 0) = 1 - p \quad \text{for each } n \ge 1.$$

Remarks

One calls $X_n = 1$ a success and $X_n = 0$ a failure in trial $n$. The number of successes in the first $n$ trials has the binomial distribution $\operatorname{Bin}(n, p)$. For $0 < p \le 1$, the number of failures before the first success has the geometric distribution with parameter $p$. The times between successes are i.i.d. The Bernoulli process is a discrete-time analogue of the Poisson process.

Explanations

L1 human

The Bernoulli process is the mathematical model of tossing the same coin again and again, forever. Each toss is independent of the others, and each has the same probability $p$ of heads.