The conditional expectation
e
X
:
R
n
R
{\displaystyle e_{X}:\mathbb {R} ^{n}\to \mathbb {R} }
is a measurable function such that
Note that unlike
X
{\displaystyle \mu _{X}}
, the conditional expectation
e
{\displaystyle e_{X}}
is not generally unique: there may be multiple minimizers of the mean squared error. However, this is a nonsensical statement: on the left is \(\textrm{E}(Y|X)\) a random variable (a function), and on the right is \(\textrm{E}(Y)\) a number. Therefore \(\textrm{E}(g(X)Y|X=x) = \textrm{E}(g(x)Y|X=x)= g(x)\textrm{E}(Y|X=x)\), where \(g(x)\) pops out of the expected value since it is just a number. Example 5. Let H be a discrete random variable taking one value from {Hit, NotHit}.
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And the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall amounts that occurred on the ten days with that specific date. 48 illustrates the ideas behind the law of total expectation and taking out what is known. 67\). Cheers!
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5 Resources To Help You Test Of Significance Based On Chi Square
To approximate \(\textrm{E}(Y|X = x)\), simulate many \((X, Y)\) pairs, discard the pairs for which \(X\neq x\), and average the \(Y\) values for the pairs that remain.
Note that conditioning on a continuous random variable is not the same as conditioning on the event
{
Y
=
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}
here are the findings
{\displaystyle \{Y=y\}}
as it was in the discrete case. .