Imagine a random variable $X$ and a function $f(x)$.
What properties must $f(x)$ have for $f(E[X]) = E[f(X)]$.
Example: Do we know the average kinetic energy of a set of particles, if we know the average speed?
Here, $f(v) = \dfrac12 m v^2$
2026-04-18 02:44:58.1776480298
What is the relationship between functions and expected values?
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$f$ should be an affine function, i.e. $f(x)=ax+b$ for some $a, b$. This follows from linearity of integration.
In general, otherwise, the equality does not hold.