Jensen's Inequality

Definition

Jensen's Inequality states that for a convex function, the expected value of the function of a random variable is greater than or equal to the function of the expected value of that random variable. This principle highlights the relationship between convex functions and expected values, emphasizing that taking the average first (the expected value) and then applying the function does not yield a result greater than applying the function first and then averaging.

5 Must Know Facts For Your Next Test

1. Jensen's Inequality is most commonly used with convex functions, but it can also be applied to concave functions with an inverted relationship.
2. If a function is linear, Jensen's Inequality becomes an equality, meaning both sides are equal when you apply the function to the expected value or vice versa.
3. Jensen's Inequality has applications in various fields such as economics, finance, and decision theory, where it helps in understanding risk and uncertainty.
4. The inequality can be represented mathematically as: if $$X$$ is a random variable and $$f$$ is a convex function, then $$f(E[X]) \leq E[f(X)]$$.
5. Understanding Jensen's Inequality helps in recognizing how risk-averse individuals might prefer certain investments due to the nature of their expected returns.

Review Questions

• How does Jensen's Inequality illustrate the impact of convex functions on expected values?
  • Jensen's Inequality shows that when dealing with convex functions, taking the expected value of a random variable first will always yield a result that is less than or equal to applying the function to the expected value. This relationship underscores how risk and uncertainty can affect decision-making. In practical terms, it means that individuals who prefer lower risk outcomes will view investments through the lens of their expected returns while considering how those returns change when affected by a convex function.
• In what situations might Jensen's Inequality lead to different interpretations when applied to concave functions versus convex functions?
  • When Jensen's Inequality is applied to concave functions, it reveals an opposite relationship compared to convex functions. In this case, applying the function before averaging yields an expected value that is less than or equal to applying the average first. This difference impacts how we interpret risk in financial contexts. For instance, risk-seeking behavior may lead individuals to favor outcomes represented by concave functions since these functions suggest higher returns on average than expected when assessed through traditional means.
• Evaluate how understanding Jensen's Inequality could affect decision-making in real-world scenarios involving risk assessment.
  • Grasping Jensen's Inequality allows individuals and organizations to make more informed decisions when faced with uncertainty and varying outcomes. By recognizing that convexity affects expectations, decision-makers can better assess investments, insurance policies, or any scenario where they face potential losses or gains. For example, an investor may choose a diversified portfolio over a single high-risk stock if they understand how expected values behave under convex functions, ultimately leading them to make choices that align with their risk tolerance and financial goals.

Related terms

Convex Function: A function is convex if, for any two points on its graph, the line segment connecting them lies above or on the graph.

Expected Value: The expected value is the long-run average value of repetitions of the experiment it represents, calculated as the sum of all possible values each multiplied by its probability.

Concave Function: A function is concave if, for any two points on its graph, the line segment connecting them lies below or on the graph.