Definition

Noun: A Bernoulli distribution is a discrete probability distribution for a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. It models a single trial of a binary experiment (e.g., success/failure, yes/no, heads/tails) where there are only two possible outcomes.

Usage

The term is used in statistics and probability theory to describe the simplest case of a discrete random variable. - The outcome of a single coin flip can be modeled using a Bernoulli distribution. - In this medical trial, whether a patient responds to the treatment is a Bernoulli distribution with success probability p.

Advanced Usage

• Parameter: The distribution is completely defined by a single parameter, , the probability of success.
• Foundation: It serves as the building block for other distributions, such as the Binomial distribution, which models the number of successes in a sequence of independent Bernoulli trials.
• Mathematical Form: If is a Bernoulli random variable, then and . Its expected value is and its variance is .

Variants and Related Words

• Bernoulli Trial: A single experiment or random event that results in a success or failure, governed by the Bernoulli distribution.
• Bernoulli Random Variable: A variable that follows a Bernoulli distribution.
• Binomial Distribution: A generalization that models the number of successes in independent Bernoulli trials.

Synonyms

• Binary Distribution
• Two-point distribution (specifically on the values 0 and 1)

Related Terms and Concepts

• Probability Mass Function (PMF): The function that gives the probability that a discrete random variable is exactly equal to some value. For a Bernoulli distribution: for .
• Indicator Variable: A Bernoulli random variable is often used as an indicator (1 if an event occurs, 0 otherwise).

Noun

1. a theoretical distribution of the number of successes in a finite set of independent trials with a constant probability of success