Independent random variables

Random variables are independent when their joint distribution equals the product of their marginal distributions. This means learning the value of one does not change the probability distribution of the others.

Probability distribution: A probability distribution specifies the possible values of a random variable and their probabilities. Independence is defined by comparing the distributions of variables jointly and separately.
Probability mass function: A function giving the probability of each possible value of a discrete random variable. For discrete variables, independence means their joint mass function factors pointwise.
Pairwise independence: A condition in which every pair among a collection of random variables is independent. Pairwise independence need not imply the mutual independence required for joint factorization.
Law of large numbers: A family of theorems stating that sample averages converge to expected values under specified conditions. Independence is a standard condition ensuring averages of repeated observations stabilize.
Copula: A multivariate distribution function that captures dependence separately from its marginal distributions. Copulas distinguish independence from other dependence structures while holding marginals fixed.
Joint probability distribution: A probability distribution describing the probabilities of combinations of values for two or more random variables. Its factorization into marginal distributions is the defining condition for independence.
Probability density function: A function whose integral over a region gives the probability that a continuous random variable lies there. For variables with a joint density, independence is equivalent to density factorization almost everywhere.
Uncorrelated random variables: Random variables are uncorrelated when their covariance is zero. Independence implies zero covariance when moments exist, but zero covariance alone does not imply independence.
Central limit theorem: A family of theorems showing that suitably standardized sums converge in distribution to a normal law. Independent summands are the classical setting for the theorem’s normal approximation.
Mutual information: A quantity measuring how much information one random variable provides about another. It is zero exactly for independent variables under standard definitions.

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