triangle inequality with random variables

by axk   Last Updated January 15, 2018 21:19 PM

Suppose $a$ and $b$ are real numbers (positive in my case), $X$ is a random variables and $F$ is a set of function. Are the following true?

  1. $|a-b| \le E_X|a-X|+E_X|X-b|$

  2. $|a-b| \le E_X [\inf_{f\in F} (|a-f(X)| + |f(X)-b|)] $

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