3
$\begingroup$

If the conditional expectation E(Z|X) is a constant k, what can be inferred about Z? Since this means that whatever the value of x is given, Z is always k, does this imply that E(Z) is equal to k?

$\endgroup$
  • $\begingroup$ You might find the Wikipedia entry on the law of total expectation useful. There are similarly useful laws for total variance and total covariance. $\endgroup$ – Glen_b Sep 29 '14 at 7:00
  • 2
    $\begingroup$ Please note that one key assertion in the question is incorrect: $Z$ is not necessarily a constant. For instance, when $Z$ and $X$ are independent, $E(Z|X)=E(Z)$ is a constant but $Z$ could have literally any distribution. On the other hand, a constant conditional expectation does not imply independence. One could start with any bivariate random variable $(X,Y)$, choose $k$, and by defining $Z=Y+k-E(Y|X)$ create $(X,Z)$ for which $E(Z|X)=k$. $X$ and $Z$ will not necessarily be independent. $\endgroup$ – whuber Sep 29 '14 at 14:34
  • $\begingroup$ @whuber Can you give an example when $X$, $Z$ are not independent in the situation you described? $\endgroup$ – wanderingmathematician Apr 7 at 17:32
6
$\begingroup$

Regarding the second part, the answer is yes $$ E(Z)=E(E(Z|X))=E(k)=k $$

$\endgroup$

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.