What is $E[X\; \mid \;Y]$ and its Law of Total Expectation?

medium

Answer

$E[X\; \mid \;Y\; = \;y]$
$E[X\; \mid \;Y]$
$E[X]\; = \;E[E[X\; \mid \;Y]]$
$(E[Y \mid X]\;\mathrm{is}\;\mathrm{the}\;\mathrm{best}\;\mathrm{predictor}\;\mathrm{of}\;Y)$

Check yourself — multiple choice

$E[X \mid Y]\; = \;0$
$E[X \mid Y = y]$ is expected X given Y=y; Adam's law: $E[X]\; = \;E[E[X \mid Y]]$
Only for discrete
Not a random variable

$E[X \mid Y]$ : conditional expectation as a function of Y; Adam's law: $E[X] = E[E[X \mid Y]]$ .

#expectation#probability

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