Probability theory MOC

Jensen’s inequality

Let (𝜉,F,ℙ) be a probability model, 𝑋 :𝜉 →ℝ be a real random variable¹, and 𝜑 :ℝ →ℝ be a Convex function. Then measure

𝜑(∫𝜉𝑋𝑑ℙ)≤∫𝜉𝜑∘𝑋𝑑ℙ𝜑(𝔼⁡[𝑋])≤𝔼⁡[𝜑(𝑋)]

with equality iff 𝜑(𝑋) =𝑎 +𝑏𝑋.

---

develop | en | SemBr

Footnotes

1. i.e. ℙ-measurable function ↩