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We have seen in the course that if X is independent of F, E[X | F] =E[X]. Is the reverse implication also correct? Namely, if we know that E[X | F] = == E[X], can we deduce that X is independent of F? This problem shows that this is NOT the case. Let X N(0, 1) be a standard normal random variable. Let Y be a random variable such that ~ - P(Y = 1) = P(Y = −1) = 1/2. Assume that X and Y are independent and let Z = XY. Let F = σ(X). Notice that X is always σ(X)-measurable and that Y is independent of F since it is independent of X. (c) Using the fact above, prove that X and Z are NOT independent. Hint: find two events A and B with AE σ(X), BE σ(Z) such that P(ANB) P(A)P(B). A typical event in σ(X) is of the form {w: X(w) < a} or {w: X(w) > a} Bonus question: Prove that Z~N(0, 1) (difficult).

We have seen in the course that if X is independent of F, E[X | F] =E[X]. Is the reverse implication also correct? Namely, if we know that E[X | F] = == E[X], can we deduce that X is independent of F? This problem shows that this is NOT the case. Let X N(0, 1) be a standard normal random variable. Let Y be a random variable such that ~ - P(Y = 1) = P(Y = −1) = 1/2. Assume that X and Y are independent and let Z = XY. Let F = σ(X). Notice that X is always σ(X)-measurable and that Y is independent of F since it is independent of X. (c) Using the fact above, prove that X and Z are NOT independent. Hint: find two events A and B with AE σ(X), BE σ(Z) such that P(ANB) P(A)P(B). A typical event in σ(X) is of the form {w: X(w) < a} or {w: X(w) > a} Bonus question: Prove that Z~N(0, 1) (difficult).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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We have seen in the course that if X is independent of F, E[X | F] =E[X]. Is the reverse implication also correct? Namely, if we know that E[X | F] = == E[X], can we deduce that X is independent of F? This problem shows that this is NOT the case. Let X N(0, 1) be a standard normal random variable. Let Y be a random variable such that ~ - P(Y = 1) = P(Y = −1) = 1/2. Assume that X and Y are independent and let Z = XY. Let F = σ(X). Notice that X is always σ(X)-measurable and that Y is independent of F since it is independent of X.
Transcribed Image Text:We have seen in the course that if X is independent of F, E[X | F] =E[X]. Is the reverse implication also correct? Namely, if we know that E[X | F] = == E[X], can we deduce that X is independent of F? This problem shows that this is NOT the case. Let X N(0, 1) be a standard normal random variable. Let Y be a random variable such that ~ - P(Y = 1) = P(Y = −1) = 1/2. Assume that X and Y are independent and let Z = XY. Let F = σ(X). Notice that X is always σ(X)-measurable and that Y is independent of F since it is independent of X.
(c) Using the fact above, prove that X and Z are NOT independent. Hint: find two events A and B with AE σ(X), BE σ(Z) such that P(ANB) P(A)P(B). A typical event in σ(X) is of the form {w: X(w) < a} or {w: X(w) > a} Bonus question: Prove that Z~N(0, 1) (difficult).
Transcribed Image Text:(c) Using the fact above, prove that X and Z are NOT independent. Hint: find two events A and B with AE σ(X), BE σ(Z) such that P(ANB) P(A)P(B). A typical event in σ(X) is of the form {w: X(w) < a} or {w: X(w) > a} Bonus question: Prove that Z~N(0, 1) (difficult).
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