Jensen's Inequality
For a random variable and a convex function ,
Proposition 3.35 (Jensen's Inequality)
Let be a random variable and be a convex function. Then
Proof
Let . By the supporting-line property of convex functions, there exist with
Then pointwise, and taking expectations,
Equality case
Suppose additionally that for there are with for all and . Since , equality forces
and a non-negative random variable with zero expectation vanishes almost surely: . By the strictness property of the supporting line this is equivalent to
Related
Stated in
- Proposition 3.35 (Jensen's Inequality)§3.3 Inequalities