Under what conditions does Jensen's inequality hold with equality?

Asked Apr 09 '19 at 17:05

Active Apr 09 '19 at 18:05

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From what I understand Jensen's Inequality is

$$E(f(X))\leq f(E(X)) \;\text{ when $f$ is a concave function}$$

Are there any conditions on the distribution of $X$ (rather than the function f) under which this holds with equality: $E(f(X))=f(E(X))$? (eg. when $VAR(X)=0$, $X$ is a constant so it should hold with equality)

Also, is there any term $O$ which would make this true: $E(f(X)) = f(E(x)) + O $ (I was thinking some sort of Taylor Expansion)

I am mainly interested in the case $f(x)=ln(x)$

edited Apr 09 '19 at 18:05

asked Apr 09 '19 at 17:05

Lorenzo

2

Possible duplicate of Convexity and equality in Jensen inequality – Sri-Amirthan Theivendran Apr 09 '19 at 17:46
I am mainly trying to show that E(ln(X))=E(ln(Y)) does not imply ln(E(X))=ln(E(Y)) and was wondering under which distribitutions of X and Y this was true (my intuition was that if X and Y have different distributions the O term would be different) – Lorenzo Apr 09 '19 at 18:11

Under what conditions does Jensen's inequality hold with equality?

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