# Jensen Inequality ### TLDR: Given a convex function $f$, we have: $f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]$ In words: > The function of the expectation is less than or equal to the expectation of the function. Just watch [this video](https://www.youtube.com/watch?v=u0_X2hX6DWE), its incredible! ![](Screen%20Shot%202022-07-14%20at%209.48.18%20PM.png) ### Why does it hold? We can show that if $f$ is linear then they are equal: ![](Screen%20Shot%202022-07-14%20at%209.48.35%20PM.png) And as we shift into the convex regime, that changes: ![](Screen%20Shot%202022-07-14%20at%209.48.52%20PM.png) ### Key idea The inequality comes from the difference the convex function has with the line. ### The Inequality is related to the variance of the rv Less variance, less difference! ![](Screen%20Shot%202022-07-14%20at%209.51.23%20PM.png) More variance, more difference! ![](Screen%20Shot%202022-07-14%20at%209.52.12%20PM.png) ### The inequality is related to the curvature of the convex function More curved functions yield larger differences! ![](Screen%20Shot%202022-07-14%20at%209.52.57%20PM.png) --- Date: 20211006 Links to: [Probability MOC](Probability%20MOC.md) [Mathematics MOC](Mathematics%20MOC.md) Tags: #todo References: * [Jensen's Inequality - YouTube](https://www.youtube.com/watch?v=u0_X2hX6DWE)