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I have always taken for granted that expected value is a linear operator. For any random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this?

Also, I know that generally median $Med()$ is not a linear operator, meaning $Med(aX + bY)$ might not be equal to $a Med(X) + b Med(Y)$. Are there absolute criteria / rules when $Med$ is a linear operator, and when it is not?

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    For the median, what you want is usually called a counterexample.2010-11-09
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    I know that. What I want is a bit more - I think there are certain conditions when this is true (and not a trivial one such as when the distribution is symmetric).2010-11-09
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    For linearity of expectation, the proof will depend on how you have defined expectation. But it should be one of the first facts about expectation proved in any textbook on probability theory, and usually follows almost immediately from the definition.2010-11-09
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    I have just edited my answer to the second question.2010-11-11

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