Jensen Inequality¶

Theorem¶

Let $X$ be an integrable random variable taking values in an interval on which $g$ is convex. If $\mathbb{E}[g(X)]$ exists (in the finite-valued form, $\mathbb{E}|g(X)|<\infty$ $\infty$ ), then

\begin{aligned} \mathbb{E}[g(X)]\geq g(\mathbb{E} X). \end{aligned}

The inequality reverses for concave $g$ .

The Expectation is the weighted average in this result. Convexity compares the function evaluated at that average with the average of the function values.

Interpretation¶

For a convex function, averaging before applying the function gives no larger a value than applying it first and then averaging: $g(\mathbb{E}X)\le\mathbb{E}[g(X)]$ $E [g (X)]$ . In general an expectation cannot be moved through a nonlinear function, and convexity tells in which direction the error goes.

The convex curve g(x) = x squared with a chord between two points x1 and x2; above the mean of X the chord, giving the mean of g(X), lies above the curve value g at the mean.

In the figure $X$ takes two values $x_1$ and $x_2$ with probability $\tfrac{1}{2}$ each, so $\mathbb{E}X$ is their midpoint. The chord joins the points of the graph of $g(x)=x^2$ above $x_1$ and $x_2$ ; above $\mathbb{E}X$ its height is the average $\mathbb{E}[g(X)]$ , while the curve has height $g(\mathbb{E}X)$ . A convex graph lies below its chords, so $g(\mathbb{E}X)\le\mathbb{E}[g(X)]$ $E [g (X)]$ . For a variable with two values this is exactly the definition of convexity, with the probabilities as weights; Jensen's inequality extends it to every distribution.

With $g(x)=x^2$ $x^{2}$ , Jensen gives $(\mathbb{E}X)^2\le\mathbb{E}[X^2]$ $E [X^{2}]$ , that is $\operatorname{Var}(X)\ge0$ $0$ , as in the first Example. For the concave $\log$ the inequality reverses, $\mathbb{E}[\log X]\le\log\mathbb{E}X$ $lo g E X$ .

A common slip is to replace $\mathbb{E}[g(X)]$ by $g(\mathbb{E}X)$ for a nonlinear $g$ . For positive $X$ , $g(x)=1/x$ is convex on $(0,\infty)$ , so $\mathbb{E}[1/X]\ge1/\mathbb{E}X$ $1/ E X$ ; if $X$ is $1$ or $3$ with equal probability, these are $\tfrac{2}{3}$ and $\tfrac{1}{2}$ . This idea returns later in insurance: with a concave utility, the expected utility of an uncertain wealth is at most the utility of its expected value.

Key Results¶

For a concave function, the inequality reverses.
If $g$ is strictly convex and both sides are finite, equality holds exactly when $X$ is constant almost surely.

Examples¶

For $g(x)=x^2$ and $X\sim\operatorname{Unif}(0,1)$ $Unif (0, 1)$ , $g(\mathbb{E}X)=1/4\le\mathbb{E}[X^2]=1/3$ $1/3$ .
For $X=1$ or $3$ with equal probability, concavity of $\log$ gives $\mathbb{E}\log X=\frac{1}{2}\log3\le\log2=\log\mathbb{E}X$ $lo g E X$ .

Prerequisites¶

Sources¶

Nicky van Foreest · Probability Distributions · Week 7

Aliases: Jensen's inequality Symbols: infinity

Last updated: 20 October 2026 03:02