Conditional Expectation for Continuous Random Variables

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Conditional Expectation for Continuous Random
Variables
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This gives us the Law of Total Probability for
Expectation
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Example 7.18
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7.4 The Bivariate Normal
We begin be recalling the univariate normal. Let
? denote the standard N(0,1) density
Then the N(m,s2) density can be written as
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For the bivariate normal, we start in a similar
way. For
If ? 0, then
is the product of two univariate standard normals.
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We show that the standard bivariate density
integrates to one. First, for all ? lt 1,
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We show that the standard bivariate density
integrates to one. First, for all ? lt 1,
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We show that the standard bivariate density
integrates to one. First, for all ? lt 1,
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From
we see that X and Y are independent if and only
if they are uncorrelated (? 0) since this is
the condition for the density to factor.
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Reminder
(7.23)
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