1) Probability and Integration Flashcards
What is a σ-algebra, and what conditions must it satisfy
What is the σ-algebra generated by a collection of subsets D
The smallest σ-algebra G that contains all elements of D is called the σ-algebra generated by D. We write G = σ(D)
What is a probability measure, and what properties must it satisfy
What is a probability space
(Ω, F, P)
What is Boole’s Inequality
What happens to the probability of an increasing or decreasing sequence of events as n approaches infinity
What does it mean for a statement to be true almost surely (a.s.) in probability
A statement is said to be true almost surely if it holds with probability one (P(A) = 1)
What is a random variable
What is the indicator function of an event A
What is the expectation of the indicator function IA of an event A
What are the key properties of expectation in probability theory
What is a convex function, and how can we check convexity
What is Jensen’s Inequality
If X is integrable and g is a convex function then
E[g(X)] ≥ g(E[X])
What is the Monotone Convergence Theorem
What is the Dominated Convergence Theorem
What is the conditional expectation of a random variable given a σ-algebra
What are the key properties of conditional expectation
Describe the proof of the conditional mean formula, E [E [X | G]] = E[X]
Describe the proof that if X is independent of G, then E[X | G] = E[X]
Describe the proof of the tower property
E[E[X | G] | G1] = E[X | G1]
