A Bernoulli process is a sequence of independent and identically distributed (i.i.d.) random variables, where each random variable takes on one of two possible values, typically denoted by 0 and 1. The process is named after the Swiss mathematician Jacob Bernoulli.
In a Bernoulli process, the probability of success (value 1) is denoted by p, and the probability of failure (value 0) is denoted by q = 1 − p. Each trial in the process is a Bernoulli-distributed random variable, and the outcomes are binary with success and failure probabilities.
The probability mass function (PMF) of a Bernoulli-distributed random variable X is given by:
$$P(X=k) = \begin{cases}
p, & \text{if } k=1, \\
q, & \text{if } k=0,
\end{cases}$$
or more compactly,
P(X = k) = pk(1 − p)1 − k for k ∈ {0, 1}.
As an example, let’s consider a simple Bernoulli process of tossing a fair coin 5 times. The probability of success (getting a head) is p = 0.5, and the probability of failure (getting a tail) is q = 0.5. Each coin toss is an independent Bernoulli trial, so the entire process is a sequence of i.i.d. Bernoulli random variables:
(X1, X2, X3, X4, X5) = (H, T, H, T, T),
where H denotes heads and T denotes tails. The probability of this particular sequence occurring is given by:
$$\begin{aligned}
P(X_1=1, X_2=0, X_3=1, X_4=0, X_5=0) &= p \cdot q \cdot p \cdot q \cdot q \\
&= 0.5^3 \cdot (1-0.5)^2 \\
&= \frac{1}{32}.\end{aligned}$$
In summary, a Bernoulli process is a sequence of independent, identically-distributed random variables with binary outcomes governed by success and failure probabilities, widely used to model many stochastic phenomena in finance, such as the evolution of stock prices or credit risk events.