Statistics flashcards, generated for you.

The set of all possible outcomes of a random experiment, denoted Ω. Every outcome in the experiment must be in Ω, and outcomes are mutually exclusive.

A collection F of subsets of Ω such that: (1) Ω ∈ F, (2) if A ∈ F then A^c ∈ F, (3) if {A_n} are countably many sets in F, then ∪A_n ∈ F. Example: the Borel σ-algebra on ℝ.

(1) P(Ω) = 1, (2) P(A) ≥ 0 for all events A, (3) countable additivity: P(∪_i A_i) = Σ P(A_i) for disjoint events.

A random variable X is a measurable function from (Ω,F,P) to ℝ. Its distribution is the probability measure induced on ℝ: μ_X(B) = P(X ∈ B). The distribution describes probabilities of X's values, not the function itself.

X_n →^P X if for all ε > 0, P(|X_n - X| > ε) → 0. Almost sure convergence is X_n →^{a.s.} X if P(lim X_n = X) = 1. Almost sure convergence is stronger: a.s. implies in probability, but not vice versa.

If X_n is a monotone increasing sequence of non-negative measurable functions with X_n ↑ X, then E[X_n] ↑ E[X]. This allows interchange of limit and expectation under monotonicity.

If X_n → X almost surely and |X_n| ≤ Y for some integrable Y, then E[X_n] → E[X]. It permits taking limits inside expectations without monotonicity; essential for proving convergence of moments.

E[X|G] is a G-measurable random variable such that for all A ∈ G, E[X·1_A] = E[E[X|G]·1_A]. It is the best L²-approximation of X using G-measurable functions.

If X_1, X_2, ... are i.i.d. with mean μ and variance σ², then (1/√n)Σ(X_i - μ)/σ converges in distribution to N(0,1). Equivalently, the law of the normalized sum converges weakly to the standard normal measure.

Measures μ_n converge weakly to μ if ∫f dμ_n → ∫f dμ for all bounded continuous f. A sequence of random variables converges in distribution iff their induced probability measures converge weakly. This is the fundamental notion in probability's structure.