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Science Forum Index » Statistics - Math Forum » problem involving conditional expectation and variance...
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 Posted: Tue Jul 01, 2008 10:46 am |
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Guest
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Hi there,
I am working on the following problem.
Prove or disprove that
Var{X} >= Var[E{X|Y}]>= Var[E{X|Y^2}]
So far, I have written the obivious
Var{X}=Var[E{X|Y}]+E(Var{X|Y})>= Var[E{X|Y}]
Var{X}=Var[E{X|Y^2}]+E(Var{X|Y^2})>= Var[E{X|Y^2}]
I am not too sure what to think about the Var[E{X|Y}]>= Var[E{X|Y^2}]
inequality
I am guessing that Y^2 contains less information than Y, so that
Var[E{X|Y}]>= Var[E{X|Y^2}] may be wrong. But I would need a more
rigorous argument.
I greatly appreciate your help! |
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| Paul Rubin... |
| Posted: Wed Jul 02, 2008 7:28 am |
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Guest
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isabellesup at (no spam) fastmail.fm wrote:
Quote: I am guessing that Y^2 contains less information than Y,
Yes.
Quote: so that
Var[E{X|Y}]>= Var[E{X|Y^2}] may be wrong.
Keep in mind that you are looking at the variance of conditional
expectations, not the conditional variance of X. At one extreme,
suppose that Y is independent of X (hence contains no useful
information). Then E{X|Y} = E{X} is a constant (variance = 0). On the
other hand, suppose that X = Y, which means Y contains as much
information about X as is possible. Then E{X|Y} = Y = X has as much
variance as X does.
/Paul |
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