Let $X$ be a random variable, and let $f$ be a concave function.

Are there any known lower bounds for (or methods of lower bounding) $\mathbb{E}[f(X)]$?

Jensen's inequality only gives an upper bound on the above.

  • 3
    $\begingroup$ An obvious lower bound is $\text{E}[f(X)]\geqslant \inf_{x\in S} f(x)$ where $S$ is the support of the random variable $X$. $\endgroup$ Nov 27, 2016 at 13:03
  • 1
    $\begingroup$ That bound given by @Stijn is the best possible in general: since there is no assumption connecting $X$ to $f$, $X$ could have all its probability concentrated in an interval $[\inf f, \inf f+\epsilon)$ for arbitrarily small positive $\epsilon$. $\endgroup$
    – whuber
    Nov 27, 2016 at 15:37

2 Answers 2


I don't know if this is helpful, but for any function $f$ set $Z \equiv f(X)$. Then by Jensen's Inequality

$$E(Z^2) > [E(Z)]^2 \implies \sqrt {E(Z^2)} > \big |E(Z)\big |$$

$$\implies -\sqrt {E(Z^2)} < E(Z) < \sqrt {E(Z^2)} $$

$$ \implies -\sqrt {E([f(X)]^2)} < E[f(X)] < \sqrt {E([f(X)]^2)} $$


A lower bound for E[f(X)]? (for a concave function f) is any upper bound on the expectation E[f(X)] of a convex function f, e.g., see Edmundson-Madansky type inequalities: 1 - Edmundson, H.P. (1956): Bounds on the expectation of a convex function of a random variable. Technical report, The Rand Corporation Paper 982, Santa Monica, California; and, 2 - Madansky, A. (1959): Bounds on the expectation of a convex function of a multivariate random variable. Ann. Math. Stat. 30, 743–746; or, Dula's type bounds: 3 - Dul´a, J.H. (1992): An upper bound on the expectation of simplicial functions of multivariate random variables. Math. Program. 55, 69–80


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.