Bernoulli refers to a family of probability distributions that are used to model binary or categorical data. It is named after Swiss mathematician Jacob Bernoulli, who introduced the concept in the 18th century. The Bernoulli distribution is a discrete probability distribution with two possible outcomes: success (typically represented by 1) and failure (typically represented by 0).

The Bernoulli distribution is often used in various fields, including statistics, machine learning, and economics, to model events with only two possible outcomes. For example, it can be used to model the probability of a coin toss resulting in heads or tails, the probability of a customer making a purchase or not, or the probability of a patient responding positively or negatively to a treatment.

The distribution is characterized by a single parameter, often denoted as p, which represents the probability of success. The probability mass function (PMF) of the Bernoulli distribution is given by:

P(X = x) = p^x * (1-p)^(1-x)

where X is a random variable representing the outcome (0 or 1), and x is the observed outcome.

The expected value (mean) of a Bernoulli distribution is E(X) = p, and the variance is Var(X) = p * (1-p). The distribution is symmetric when p = 0.5 and becomes skewed towards one of the outcomes as p deviates from 0.5.

The Bernoulli distribution is a special case of the binomial distribution, which models the number of successes in a fixed number of independent Bernoulli trials.