5) Expectation

Question 1

What is the Expectation of a random variable X defined on (Ω,F,P)

Answer 1

Question 2

What are the properties of Expectation

Answer 2

Question 3

How is the Lebesgue-Stieltjes (LS) integral constructed for X≥0 when X is a simple/elementary random variable

Answer 3

Question 4

How is the Lebesgue-Stieltjes (LS) integral constructed for X≥0 when X is a general random variable

Answer 4

Question 5

How is the Lebesgue-Stieltjes (LS) integral constructed for X∈R, a general random variable

Answer 5

Question 6

What does it mean for a random variable X to be integrable

Answer 6

Question 7

How is the Lebesgue-Stieltjes (LS) integral constructed with respect to a general measure μ

Answer 7

Question 8

How is the LS integral written when S=R and 𝜇=λF

Answer 8

Question 9

What is the “standard machine” in the context of proving linear statements

Answer 9

Question 10

What is the Monotone Convergence Theorem

Answer 10

Question 11

What is Fatou’s lemma

Answer 11

Question 12

What is the Dominated Convergence Theorem

Answer 12

Question 13

Why is the dominance condition necessary in the Dominated Convergence Theorem

Answer 13

Question 14

Why is the dominance condition necessary in the Dominated Convergence Theorem, and what happens if it is removed

Answer 14

Question 15

What is Scheffe’s Lemma

Answer 15

Question 16

Is the dominance condition necessary for the Dominated Convergence Theorem, and how does it relate to Scheffé’s Lemma

Answer 16

Question 17

What is the relationship between null sets and expectation

Answer 17

Question 18

How do null sets affect the assumptions in key results like the Monotone Convergence Theorem and Dominated Convergence Theorem

Answer 18

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.