BernoulliExperimente

BernoulliExperimente are a fundamental concept in probability theory used to model simple yes-or-no experiments. Each Bernoulli experiment has exactly two possible outcomes, commonly labeled success and failure. The probability of success is a fixed value p, with 0 ≤ p ≤ 1, and the probability of failure is 1 − p. In a single trial, the outcome is determined randomly according to these probabilities.

When a Bernoulli experiment is repeated n times under the same conditions in a sequence of independent trials, the number of successes follows a binomial distribution with parameters n and p. This connection underpins many statistical methods, including estimations, hypothesis testing, and confidence intervals. The Bernoulli distribution itself describes the outcome of a single trial, where the random variable takes the value 1 with probability p and 0 with probability 1 − p.

Key properties include the mean and variance of a Bernoulli random variable X: E[X] = p and Var(X) = p(1 − p). For a sequence of n independent Bernoulli(p) trials, the sum S of the outcomes has a Binomial(n, p) distribution, with E[S] = np and Var(S) = np(1 − p).

Bernoulli experiments are widely used to model simple yes/no events in fields such as quality control, genetics, survey sampling, and A/B testing. They are named after Jacob Bernoulli and form the basis for broader concepts like Bernoulli processes and binomial distributions.