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Is there any mathematical association between the conditional expectation of a variable given another variable, and the unconditional expectation of that variable?

I realise that given a joint distribution between $A$ and $B$, we can find $E[A]$ by first finding the marginal distribution of $A$.

However, suppose we know what is $E[A|B]$, is it possible to obtain $E[A]$ from this alone using any mathematical property?

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No. You also need additional information to obtain $\mathrm{E}(A)$. In particular, the probability distribution of $B$ helps. We can write

$$ \mathrm{E}(A) = \sum_b \mathrm{E}(A|B=b) \mathrm{P}(B=b) $$

This means that if you know $\mathrm{E}(A|B)$ and $\mathrm{P}(B)$, then you can calculate $\mathrm{E}(A)$.

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  • $\begingroup$ The answer of Kota Mori is perfect but note that what was written is also referred to as the tower property of expectations: E(E(A|B)) = E(A). $\endgroup$
    – mlofton
    Oct 28, 2020 at 13:04
  • $\begingroup$ @mlofton Thanks. Also called "law of total expectation". en.wikipedia.org/wiki/Law_of_total_expectation $\endgroup$
    – Kota Mori
    Oct 28, 2020 at 13:51

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