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3. Fix a probability space (2, F, P). Let X and Y be two independent continuous random variables with joint density function px,y(x, y) and marginal density functions px(x), PY (y). Let g(X, Y) be a function with E[ lg(X,Y)]]<∞ and we define f(y) := E[g(x, y)]. Show that E[g(X,Y) | Y] = f(Y).

3. Fix a probability space (2, F, P). Let X and Y be two independent continuous random variables with joint density function px,y(x, y) and marginal density functions px(x), PY (y). Let g(X, Y) be a function with E[ lg(X,Y)]]<∞ and we define f(y) := E[g(x, y)]. Show that E[g(X,Y) | Y] = f(Y).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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3. Fix a probability space (2, F, P). Let X and Y be two independent continuous random variables with joint density function px,y(x, y) and marginal density functions px(x), PY (y). Let g(X, Y) be a function with E[ lg(X,Y)]]<∞ and we define f(y) := E[g(x, y)]. Show that E[g(X,Y) | Y] = f(Y). Hint: Use Fubini's theorem; also, the proof of Example 1.16 (ii) in the lecture notes may be helpful. Hint: Recall that for a random variable Z that is o(Y)-measurable, there exists some measurable function h such that Z= h(Y). You may use this result without proof. Remark. This problem has a more general version: Let G be a sub-sigma field of F. Assume that X is independent from G and that Y is G-measurable. Can you show that E[g(X, Y) | Y] = f(Y)?
Transcribed Image Text:3. Fix a probability space (2, F, P). Let X and Y be two independent continuous random variables with joint density function px,y(x, y) and marginal density functions px(x), PY (y). Let g(X, Y) be a function with E[ lg(X,Y)]]<∞ and we define f(y) := E[g(x, y)]. Show that E[g(X,Y) | Y] = f(Y). Hint: Use Fubini's theorem; also, the proof of Example 1.16 (ii) in the lecture notes may be helpful. Hint: Recall that for a random variable Z that is o(Y)-measurable, there exists some measurable function h such that Z= h(Y). You may use this result without proof. Remark. This problem has a more general version: Let G be a sub-sigma field of F. Assume that X is independent from G and that Y is G-measurable. Can you show that E[g(X, Y) | Y] = f(Y)?
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