double conditional expectation

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Isabelle - 16 Feb 2007 17:53 GMT

I am trying to prove that for bounded X and Y,
E{Y E[X| F ]}= E{X E[Y| F ]}
where F is a sigma-field

Shall I try to prove the statement with X and Y indicator variables first? Then, combinations of indicator variables? Then make the transition to variables X and Y by a convergence theorem?

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ArtflDodgr - 16 Feb 2007 20:01 GMT

> I am trying to prove that for bounded X and Y,
> E{Y E[X| F ]}= E{X E[Y| F ]}
[quoted text clipped - 3 lines]
> Then, combinations of indicator variables? Then make the transition to
> variables X and Y by a convergence theorem?

More or less directly from the definition of conditional expectation,
you have the following: If A and B are bounded random variables and if A
is F-measurable, then E[AB] = E{A E[B|F]}.

Use this twice, first with A = E[X|F] and B = Y, and then with
A = E[Y|F} and B = X, to see that the two sides of your identity-to-be
are both equal to E{E[X|F] E[Y|F]}.

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Math Forum / Mathematics / Research / February 2007

double conditional expectation