5) Expectation Flashcards
What is the Expectation of a random variable X defined on (Ω,F,P)
What are the properties of Expectation
How is the Lebesgue-Stieltjes (LS) integral constructed for X≥0 when X is a simple/elementary random variable
How is the Lebesgue-Stieltjes (LS) integral constructed for X≥0 when X is a general random variable
How is the Lebesgue-Stieltjes (LS) integral constructed for X∈R, a general random variable
What does it mean for a random variable X to be integrable
How is the Lebesgue-Stieltjes (LS) integral constructed with respect to a general measure μ
How is the LS integral written when S=R and 𝜇=λF
What is the “standard machine” in the context of proving linear statements
What is the Monotone Convergence Theorem
What is Fatou’s lemma
What is the Dominated Convergence Theorem
Why is the dominance condition necessary in the Dominated Convergence Theorem
Why is the dominance condition necessary in the Dominated Convergence Theorem, and what happens if it is removed
What is Scheffe’s Lemma
Is the dominance condition necessary for the Dominated Convergence Theorem, and how does it relate to Scheffé’s Lemma
What is the relationship between null sets and expectation
How do null sets affect the assumptions in key results like the Monotone Convergence Theorem and Dominated Convergence Theorem
The assumptions in the Monotone Convergence Theorem (0≤Xn↑X), Fatou’s Lemma, the Dominated Convergence Theorem (Xn →X and ∣Xn∣≤Z), and Scheffé’s Lemma (Xn →X) need only hold almost surely (P-a.s.), meaning they are valid on a set of F-measurable outcomes with P-measure one. This highlights that null sets (sets with P(A)=0) do not impact these results.
