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 May 18 awarded Supporter May 18 comment Computing real integrals using the Residue Theorem where singularities are on the real line You have $\int_0^\inf \frac{\sin(x)/x}{x^2+a^2}$. The function $\sin(x)/x$ does not have a pole at $x=0$ (strictly speaking it's undefined at $x=0$ but it has a holomorphic extension that includes the origin). So there are only two poles, $ia$ and $-ia$, both of which are not on the real line. May 4 awarded Student Sep 11 revised Under what conditions is expectation value distributive? added 282 characters in body Sep 11 comment Under what conditions is expectation value distributive? Thanks to everyone who tried to answer my question. I have now found a partial answer to my question. Unfortunately there is no guarantee that given marginal or conditional distributions for $X,Y$ and $X,Z$, there exists a joint distribution for $X,Y,Z$. This is why the distributive law does not always hold. The paper dealing with conditional distributions can be seen here. However, if anyone can lucidate the matter with a simple condition, the question is still open. I have also editted the question to make it clearer what I want. Sep 10 comment Under what conditions is expectation value distributive? Thanks for your answer Michael. Here is what I don't understand. Suppose $X,Y$ follow some measure $dp(x,y)$ (where $x,y$ are particular values of $X,Y$) and $X,Z$ follow $dq(x,y)$, where you can assume that $p$ and $q$ are densities. Now of course $\int dy p(x,y) = \int dz q(x,z)$. It is not obvious to me that $E(XY) + E(XZ) = int dx dy p(x,y) x y + \int dx dz q(x,z) x z = \int dx dy dz x(y+z) r(x,y,z) = E\bigl(X(Y+Z)\bigr)$. Does such extension of $p$ and $q$ always exist? Sep 10 awarded Editor Sep 10 comment Under what conditions is expectation value distributive? Thanks mjqxxxx, editted. I think it can't always be true because the violation of this property is needed to arrive at the violation of Bell inequalities in quantum mechanics. Sep 10 revised Under what conditions is expectation value distributive? edited title Sep 10 asked Under what conditions is expectation value distributive?