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Suppose a random vector in $\mathbb{R}^n$ has a continuous distribution over a convex set. I want a simple proof that the expected value of the random vector will also lie in this set.

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The expected value is the integral of vector $x$ over the set with respect to the distribution, so is the limit of finite convex combinations of elements of the set, hence is in the (closed) set. If the set isn't closed you have to work a little harder, but not much.

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