Bernoulliformuleringen

Bernoulliformuleringen refers to the mathematical formulation of Bernoulli trials and the Bernoulli distribution, a foundational concept in probability theory for binary outcomes. In this formulation a Bernoulli random variable X assumes the value 1 (success) with probability p and the value 0 (failure) with probability 1-p. The probability mass function is P(X=1)=p and P(X=0)=1-p.

Key properties include the expectation E[X] = p and the variance Var(X) = p(1-p). If X1, X2, ..., Xn are independent Bernoulli(p) variables, their sum S = X1 + X2 + ... + Xn follows a Binomial(n,p). The probability of exactly k successes is P(S=k) = C(n,k) p^k (1-p)^{n-k}. A sequence of independent Bernoulli trials with constant p is called a Bernoulli process. The probability generating function for X is G(t) = E[t^X] = (1-p) + p t; for the sum S it is G(t)^n.

Estimation and usage: The maximum likelihood estimate of p from a sample is p_hat = (1/n) sum Xi, with variance p(1-p)/n. Bernoulli formulations are widely used to model binary outcomes in quality control, clinical trials, surveys, reliability studies, and risk assessment.

Related distributions include the geometric distribution, which describes the number of trials until the first success, and the negative binomial distribution, which describes the number of trials until r successes.

History: The Bernoulli distribution is named after Jacob Bernoulli, and it forms the simplest nontrivial discrete distribution, serving as a building block for more complex probabilistic models.