Probability flashcards, generated for you.

A triple (Ω, ℱ, P) where Ω is the sample space, ℱ is a σ-algebra of events, and P is a probability measure satisfying P(Ω)=1, P(∅)=0, and countable additivity.

A collection ℱ of subsets of Ω that is closed under complements and countable unions: (1) Ω ∈ ℱ, (2) A ∈ ℱ ⟹ Aᶜ ∈ ℱ, (3) {Aₙ} ⊆ ℱ ⟹ ⋃Aₙ ∈ ℱ.

Countable additivity ensures consistency with limits and enables rigorous treatment of continuous probability distributions; finite additivity alone cannot guarantee P(⋃ Aₙ) = Σ P(Aₙ) for infinite sequences of disjoint events.

A measurable function X: Ω → ℝ such that for all Borel sets B ⊆ ℝ, the preimage X⁻¹(B) = {ω ∈ Ω : X(ω) ∈ B} belongs to ℱ.

The pushforward measure μₓ on ℝ defined by μₓ(B) = P(X ∈ B) = P(X⁻¹(B)) for all Borel sets B; it completely characterizes the probabilistic behavior of X.

For a random variable X: Ω → ℝ, E[X] = ∫_Ω X dP; if X ≥ 0 or E[|X|] < ∞, this Lebesgue integral represents the expectation, equivalent to ∫_{-∞}^{∞} x dμₓ(x).

Events A, B ∈ ℱ are independent if P(A ∩ B) = P(A)P(B); random variables X, Y are independent if σ(X) and σ(Y) are independent sub-σ-algebras, i.e., P(X ∈ A, Y ∈ B) = P(X ∈ A)P(Y ∈ B) for all Borel A, B.

If {Xₙ} is a sequence of random variables with Xₙ ↑ X almost surely (or monotone increasing), then E[Xₙ] → E[X], even if the limit expectation is infinite.

A ℊ-measurable random variable satisfying ∫_G X dP = ∫_G E[X|ℊ] dP for all G ∈ ℊ; it is the orthogonal projection of X onto the closed subspace L²(Ω, ℊ, P) in the Hilbert space L²(Ω, ℱ, P).

The Borel σ-algebra on ℝ is generated by open intervals and is minimal; it allows Lebesgue measure to be defined canonically on all 'reasonable' subsets, enabling probability distributions (like normal, exponential) to be formalized via σ-finite measures satisfying regularity conditions.