Bernoullivariabel

Bernoullivariabel, commonly called a Bernoulli random variable, is the simplest discrete probability model for a binary outcome. It takes two values, typically 0 and 1, with a single parameter p in [0,1] that specifies the probability of the value 1 (success). The probability mass function is P(X=1)=p and P(X=0)=1−p.

Key statistics describe its behavior. The expectation is E[X]=p and the variance is Var(X)=p(1−p). The moment generating function is M_X(t)=1−p+p e^t, and the characteristic function is φ_X(t)=1−p+p e^{it}. These expressions enable straightforward calculations of moments and distributions of functions of X.

Interpretation and use: A Bernoulli variable models a single Bernoulli trial, such as a coin flip, pass/fail, or yes/no decision. It provides a fundamental building block for more complex distributions and models binary data.

Relationship to other distributions: If X1,...,Xn are independent Bernoulli(p) variables, their sum S=∑ Xi has a Binomial(n,p) distribution. The Bernoulli distribution is thus the n=1 instance of the binomial family.

Special cases and notation: If p=0 or p=1, the variable is degenerate, taking a constant value. The standard notation is X ~ Bernoulli(p). In many languages the term is rendered as Bernoullivariabel or Bernoulli-variabel, but the meaning remains the same.

Bernoulli variables underpin many probabilistic models and appear across statistics, machine learning, and quality control as a model for binary outcomes.