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Conditional expectation

{title2=$\mathbb E[\mathbb E[X\mid\mathcal G]]=\mathbb E[X]$}
{wiki}

Taking expectation after conditioning recovers the original expectation. More generally, if $\mathcal H\subseteq\mathcal G$, then
$$
\mathbb E[\mathbb E[X\mid\mathcal G]\mid\mathcal H]
=\mathbb E[X\mid\mathcal H].
$$

= Tower property of conditional expectation
{synonym}

= Law of iterated expectation
{synonym}


Law of total expectation

codex/tower-property-of-conditional-expectation

codex/law-of-iterated-expectation

Law of total expectation ( $E [E [X ∣ G]] = E [X]$ )

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Law of total expectation (E[E[X∣G]]=E[X])

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Law of total expectation ( $E [E [X ∣ G]] = E [X]$ )

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