Quiz: σ-Algebras and Probability Spaces
Module 1 of 5 · Easy
Quick Quiz
1. Let Ω = {a, b, c}. Which of the following collections is a valid σ-algebra on Ω?
2. The Borel σ-algebra B(ℝ) is generated by the open sets of ℝ. Which of the following is an equivalent generating family?
3. Let Ω = {1, 2, 3, 4} and X: Ω → ℝ with X(1) = X(2) = 0 and X(3) = X(4) = 1. What is σ(X), the σ-algebra generated by X?
4. A collection F on Ω is closed under finite unions but NOT necessarily under countable unions (all other σ-algebra axioms hold). This structure is called an algebra. What is the key additional requirement that upgrades an algebra to a σ-algebra?
5. Under probability measure P on Ω = {H, T} with P({H}) = 0.6 and P({T}) = 0.4, random variable X maps H ↦ 2 and T ↦ −1. What is E_P[X]?
6. Why is it impossible to define a translation-invariant probability measure on ALL subsets of [0, 1]?
7. Which statement about completeness of a probability space is correct?
8. A structuring quant says: 'Our payoff depends on the closing price S_T and on whether the stock ever touched barrier H during [0, T].' The payoff must be measurable with respect to which σ-algebra?