Probability flashcards, generated for you.

A function P: F → [0,1] where F is a σ-algebra on Ω, satisfying: (1) P(Ω)=1, (2) P(∅)=0, (3) countable additivity—if {Aₙ} are pairwise disjoint sets in F, then P(⋃Aₙ)=ΣP(Aₙ).

A collection F of subsets of Ω closed under complements and countable unions. It's necessary because probability must handle infinite sequences of events (countable unions/intersections), which raw power sets cannot always accommodate measurably.

P(A|B) = P(A∩B)/P(B), defined only when P(B)>0. It represents the probability of A given that B has occurred, updated from the prior probability P(A).

If {Bₙ} is a countable partition of Ω (pairwise disjoint, ⋃Bₙ=Ω), then P(A) = ΣP(A|Bₙ)P(Bₙ). This decomposes probability over disjoint cases.

A and B are independent if P(A∩B) = P(A)P(B). Equivalently, P(A|B)=P(A) when P(B)>0. Conditional information about B does not alter the probability of A.

A random variable X: Ω → ℝ is a measurable function—for every Borel set B⊆ℝ, the preimage X⁻¹(B) is in the σ-algebra F. Measurability ensures P(X∈B) is well-defined.

F_X(x) = P(X ≤ x). Properties: (1) non-decreasing, (2) right-continuous, (3) lim_{x→-∞}F_X(x)=0, lim_{x→+∞}F_X(x)=1. The CDF uniquely determines the distribution of X.

Discrete: p(x)=P(X=x)>0 for countably many x; P(X∈A)=Σp(x). Continuous: f(x)≥0 satisfies ∫f dx=1; P(X∈A)=∫_A f dx. For continuous X, P(X=x)=0 for any single x.

E[X] = ∫Ω X dP = ∫_{-∞}^{∞} x dF_X(x). It exists if E[|X|]<∞ (absolute integrability). For discrete: E[X]=Σx·p(x); for continuous: E[X]=∫x·f(x)dx.

If X_n→X almost surely and |X_n|≤Y with E[Y]<∞, then E[X_n]→E[X]. It justifies interchanging limits and expectations, essential for proving consistency of estimators and convergence of moments in real analysis of probability.