verteilenverteilt

Verteilenverteilt is a term occasionally used in probability theory and statistics to describe a distribution that arises when the parameter controlling the distribution is itself random. It captures a two-stage stochastic process: first, a parameter θ is drawn from a distribution G, and then a random variable X is drawn from a family Fθ parameterized by θ. The resulting marginal distribution of X is obtained by integrating over θ, effectively distributing one distribution over another.

Formal definition: Let θ be a random parameter with distribution G, and conditional on θ, X follows the distribution Fθ. The marginal distribution of X is F_X(x) = ∫ Fθ(x) dG(θ). If θ takes discrete values, this becomes a finite mixture; if θ is continuous, it is a continuous mixture. Moments of X satisfy E[X] = Eθ[E[X|θ]] and Var[X] = Eθ[Var(X|θ)] + Varθ[E[X|θ]].

Examples: A classic normal example is X|μ ~ N(μ, σ^2) with μ ~ N(μ0, τ^2); the marginal distribution of X is normal with mean μ0 and variance σ^2 + τ^2. Another important case is Poisson with a Gamma mixing distribution: X|λ ~ Poisson(λ) and λ ~ Gamma(k, θ), which yields a Negative Binomial distribution for X.

Relation to other concepts: Verteilenverteilt overlaps with mixture distributions and Bayesian hierarchical models. It emphasizes the randomness of the parameter and the induced compound or hierarchical structure in the resulting distribution.

Terminology note: The exact term "verteilenverteilt" is not standard across all literature; more common expressions include mixture distribution, hierarchical model, or random-parameter (or compound) distributions.