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Example Probability flashcards
What is a probability space?
A triple (Ω, ℱ, P) where Ω is the sample space, ℱ is a σ-algebra of subsets of Ω, and P is a probability measure satisfying P(Ω)=1 and countable additivity.
What is a σ-algebra and why is it necessary for probability?
A collection ℱ of subsets closed under complements and countable unions. It ensures P is well-defined and countably additive, avoiding pathological non-measurable sets.
State the axioms of probability measure P.
Non-negativity: P(A)≥0 for all A∈ℱ; Normalization: P(Ω)=1; Countable Additivity: P(⋃ᵢ Aᵢ)=Σᵢ P(Aᵢ) for disjoint sequences in ℱ.
What is conditional probability and what requirement must hold?
P(A|B) = P(A∩B)/P(B), defined when P(B)>0. It satisfies the axioms on the restricted space and provides the basis for Bayes' theorem.
State the law of total probability.
If {Bᵢ} is a countable partition of Ω with P(Bᵢ)>0 for all i, then P(A) = Σᵢ P(A|Bᵢ)P(Bᵢ) for any event A.
What does independence mean for events and how does it extend to σ-algebras?
Events A, B are independent if P(A∩B)=P(A)P(B). σ-algebras ℱ, 𝒢 are independent if P(F∩G)=P(F)P(G) for all F∈ℱ, G∈𝒢.
Define a random variable and its induced measure.
A measurable function X: Ω→ℝ where X⁻¹(B)∈ℱ for all Borel B⊂ℝ. The induced measure μ_X on ℝ is μ_X(B)=P(X⁻¹(B)), called the distribution of X.
What is the relationship between expectation and the Lebesgue integral?
The expectation E[X] = ∫_Ω X dP is the Lebesgue integral of X with respect to P. It equals ∫_ℝ x dμ_X(x) by the change-of-variables formula.
State the Monotone Convergence Theorem for probability.
If Xₙ is an increasing sequence of non-negative random variables with Xₙ→X almost surely, then E[Xₙ]→E[X], allowing limit and expectation to commute.
What does the Borel-Cantelli Lemma assert and when is the converse true?
If Σₙ P(Aₙ)<∞, then P(lim sup Aₙ)=0. The converse holds when {Aₙ} are independent: if Σₙ P(Aₙ)=∞ and independent, then P(lim sup Aₙ)=1.
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