LaTeX
f(\mathbb E[X])\leq\mathbb E[f(X)]Variables
- f: convex function
- X: random variable
- E: expectation
How to use this formula
Relates a convex function of an expectation to the expectation of the convex function.
Important notes
- The inequality reverses for concave f.
- Integrability and domain conditions must hold.
Quick example
For f(x)=x², (E[X])²≤E[X²].
Applicability, worked calculation, and verification
Assumptions and domain checks
- The inequality reverses for concave f.
- Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result.
Worked example
For f(x)=x², (E[X])²≤E[X²].
Common mistakes
- Before substituting values into Jensen’s Inequality, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions.
Continue the workflow
Use Jensen’s Inequality in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
f(\mathbb E[X])\leq\mathbb E[f(X)]. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Jensen’s Inequality used for?
Relates a convex function of an expectation to the expectation of the convex function.
Can I copy this formula as LaTeX?
Yes. Copy f(\mathbb E[X])\leq\mathbb E[f(X)] or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.