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Jensen for logarithm and exponential

Example

Let $X$ be a positive integrable random variable such that $\log X$ is integrable.

Applying Jensen to the concave function $\log$ yields $E [\log X] \leq \log E [X],$ equivalently $\exp (E [\log X]) \leq E [X] .$
If $Y$ is any integrable real random variable with $E [e^{Y}] < \infty$ , then Jensen applied to the convex function $e^{t}$ gives $e^{E [Y]} \leq E [e^{Y}] .$

For a two-point law with $P (X = a) = p$ and $P (X = b) = 1 - p$ , $a, b > 0$ , the first inequality is the weighted arithmetic-geometric mean inequality $a^{p} b^{1 - p} \leq p a + (1 - p) b .$

Facts & Assumptions

Given: A positive integrable random variable $X$ such that $\log X$ is integrable, and an integrable real random variable $Y$ with finite exponential moment.

[L1]

Jensen's inequality holds for expectation under the stated integrability hypotheses (Jensen's inequality for expectation).

Verification

technique · direct

1.1

Apply [L1] to the convex function $t \mapsto e^{t}$ and to $Y$ . This gives $e^{E [Y]} \leq E [e^{Y}] .$

1.2

Applying the same theorem to $Z = \log X$ and the convex function $t \mapsto e^{t}$ gives $e^{E [\log X]} \leq E [e^{\log X}] = E [X],$ which is equivalent to $E [\log X] \leq \log E [X]$ .

L1algebra

2.1

For the two-point law, step 1.2 becomes $e^{p \log a + (1 - p) \log b} \leq p a + (1 - p) b,$ that is, $a^{p} b^{1 - p} \leq p a + (1 - p) b .$

step 1.2algebra

3.1

Steps 1.1, 1.2, and 2.1 give the exponential-moment and weighted AM-GM forms of Jensen.

step 1.1step 1.2step 2.1∎

Depends on

Jensen's inequality for expectation

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)