0
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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 $\mathrm{E}(A)$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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 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 at 13:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.