0

What is the name of the property of some function, $G(x)$, such that $\mathbb{E}[G(x)]=G(\mathbb{E}[x])$? Thank you.

StubbornAtom
  • 17,932
Charles
  • 705
  • I am not aware if this is a named property. You could say $G,\mathbb E$ commute. Note that this property is always true if $G(x)$ is of the form $ax+b,a,b\in\mathbb R$ i.e. $G(x)$ is linear. – Shubham Johri Oct 21 '20 at 10:01
  • @StubbornAtom Yes, thank you. – Charles Oct 21 '20 at 17:22

1 Answers1

1

Using Jensen's Inequality the statement is true ONLY if $g$ is linear.

In fact, for $g$ convex,

$$g(\mathbb{E}[X]) \leq\mathbb{E}[g(X)] $$

tommik
  • 33,201
  • 4
  • 17
  • 35
  • Equality in Jensen's inequality holds, according to the article, iff $X$ is constant a.s. or $g$ is a.s. equal to an affine function, but even that doesn't answer the question as it assumes $g$ is convex. Does the equality $g(\mathbb{E}[X])=\mathbb{E}[g(X)]$ already imply $g$ is convex? – Thorgott Oct 21 '20 at 16:14