I'll be using the symbols from my writeup on the Jensen inequality, so you'll want to read that first. These proof of ... writeups all start the same, don't they? I get so bored, talking to myself and nobody ever *listens*. I mean, I could just write EDB borg `x`^{y}=17 probability and people would still softlink it to insulting titles but still vote it up and C! it.

OK, so let's write `x` = **E**X. Being convex, the function `f` has a supporting line at `x`. That is, there exists a linear function `L`(`t`) = a`t`+b for which `L(x)`=`f(x)` and `L(t)` ≤ `f(t)` for all `t` ("`f` remains above `L`").

The simplicity of the truth of the inequality is revealed!

**E**`f`(X) ≥ **E**`L(X)` =
b + a**E**X =

b + a`x` = `L(x)` = `f`(`x`) = `f`(**E**X)

Q.E.D.