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)
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Chapter1: Combinatorial Analysis
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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)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe591e3e-c1e9-4180-ad99-642a1f427277%2F0baf06ad-a3f9-4ba6-9634-548450d7f11d%2Feytc2iqc_processed.jpeg&w=3840&q=75)
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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