How to prove that exponential of an expected value of a variable is less than the expected value of the exponential of the variable

Question

I am trying to prove the following:

$e^{E(x)} \le E(e^x)$ for a discrete random variable x.

I am stuck on how to proceed. None of the usual rules for expected value seem to apply for something like $f(E(x))$. Can I some help? Thanks!!

Have you ever heard of Jensen's inequality( https://en.wikipedia.org/wiki/Jensen%27s_inequality)? — Seewoo Lee, Nov 24 '18 at 03:16

score 0 · Accepted Answer · answered Nov 24 '18 at 03:18

0

Exponent is a convex function. Use, as suggested in the comments, Jensen's inequality

answered Nov 24 '18 at 03:18

Makina

1 Answers1