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Jensen's inequality for expectation

Statement

Let $(Ω, F, P)$ be a probability space, let $X$ be an integrable real random variable, let $I \subseteq R$ be an interval containing $X (ω)$ for almost every $ω$ , and let $φ : I \to R$ be convex.

Assume additionally that $φ (X)$ is either integrable or nonnegative, so its expectation is defined by Expectation of a nonnegative or integrable random variable. Then $φ (E [X]) \leq E [φ (X)] .$ In the nonnegative case the right-hand side may be $+ \infty$ .

Facts & Assumptions

Given: A probability space, an integrable real random variable $X$ , an interval $I$ containing its almost-everywhere range, and a convex $φ : I \to R$ such that $φ (X)$ is integrable or nonnegative.

[L1]

Expectation is integration against the underlying probability measure (Expectation of a nonnegative or integrable random variable).

[L2]

Jensen's integral inequality holds for a probability measure whenever the composed function is integrable (Jensen's integral inequality for a probability measure).

Proof

technique · direct

1.1

If $φ (X)$ is nonnegative and $E [φ (X)] = + \infty$ , then the claimed inequality is automatic, because $φ (E [X])$ is a real number while the right-hand side is $+ \infty$ .

given

1.2

In every remaining case, $φ (X)$ is integrable: this is assumed directly, or follows from nonnegativity and finite expectation. Thus [L1] and [L2] applied to the probability measure $P$ give $φ (E [X]) = φ (\int X d P) \leq \int φ (X) d P = E [φ (X)] .$

L1L2

2.1

Steps 1.1 and 1.2 cover the infinite and finite expectation cases.

step 1.1step 1.2∎

Depends on

Used by

Jensen for logarithm and exponential Example

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.6.1 (standard reference, not scraped)
J. R. Norris, Probability and Measure, Theorem 4.3.2 (standard reference, not scraped)