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Quiz 1 Questions
February 21, 2023
MATH3801 submit only question 1. MATH3901/5901 submit question 1 and
question 2. Students should write their student # and course code in
their uploaded (preferably typeset) submission.
1. Suppose we have the probability space (Ω,H,P) and the positive random
variable X ≥ 0 with 0 < EX = ∫ XdP < ∞. Define the set function
Q : H 7→ R via:
Q(A) :=
E[X1A]
E[X]
, A ∈ H.
Using the Monotone Convergence Theorem (if Yn ↑ Y , then EYn ↑ EY ),
prove that Q is a probability measure.
2. Suppose that the following conditions are met for the collection of random
variables X, Y, (Xn), (Yn) as n ↑ ∞:
(a) Xn → X;
(b) |Xn| ≤ Yn;
(c) Yn → Y and EYn → EY <∞.
Using Fatou’s Lemma on Yn ±Xn, prove that EXn → EX.