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What is Jensen's Inequality?
Jensen's Inequality is a fundamental result in mathematical analysis that relates the convexity of a function to its expected value. It provides a mathematical relationship that holds for convex or concave functions and is widely used in various fields, including economics, finance, and probability theory. In economics and finance, it is used to establish bounds on risk and return relationships, pricing models, portfolio optimization, and utility theory. In probability theory, it helps derive bounds on expected values and moments of random variables.
Jensen's Inequality applies to convex functions, but what is a convex function?
A function is considered convex if the line segment connecting any two points on the graph lies above or on the graph. Intuitively, it means the function curves upward or is "bowed" upward. Geometrically, Jensen's Inequality can be visualized as the fact that the graph of a convex function lies below the secant line connecting two points on the graph. The inequality implies that the average value of a convex function is greater than or equal to the function evaluated at the average value.
Jensen's Inequality states that for a convex function, the expected value of the function evaluated at a random variable is greater than or equal to the function of the expected value of the random variable. Symbolically, for a convex function f(x) and a random variable X, it can be expressed as E[f(X)] ≥ f(E[X]). The expected value E[X] of a random variable X represents the average value it would take over a large number of trials. In the context of Jensen's Inequality, the expected value is calculated by summing or integrating the product of the random variable's values and their respective probabilities or densities.
Jensen's Inequality also holds for concave functions, but with the direction of the inequality reversed. For a concave function, the expected value of the function evaluated at a random variable is less than or equal to the function of the expected value. Symbolically, for a concave function f(x) and a random variable X, it can be expressed as E[f(X)] ≤ f(E[X]).