# Conditional Expectation Constant

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?

• You might find the Wikipedia entry on the law of total expectation useful. There are similarly useful laws for total variance and total covariance. Sep 29 '14 at 7:00
• 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.
– whuber
Sep 29 '14 at 14:34
• @whuber Can you give an example when $X$, $Z$ are not independent in the situation you described? Apr 7 '19 at 17:32

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