I have a discrete random variable $X$ and I know its expectation $\bar{X} = \mathbb{E}[X] \in \mathbb{R}$. I would like to say that the minimum of the support of $X$ must be $\leq \bar{X}$. Assuming this claim is correct, I couldn't see any problems with it, I would like to know if there's a general theorem out there I should be referencing when making this claim. Maybe something about $\bar{X}$ existing in the convex hull of the support of $X$?
2026-03-28 00:54:19.1774659259
Theorem Bounding Minimum Supported Value Given An Expectation of a Discrete Random Variable
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