In general, the function of the expectation is not equal to the [[Expectation]] of a function. This only only the case for [[Linear Functions of Random Variables]]. $ g(\mathbb E[X]) \neq \mathbb E[g(X)]$ For example: $ g(X)= X^2: \big(\mathbb E[X] \big)^2 \neq \mathbb E[X^2]$ However, for [[Identify Convex and Concave Functions|convex and concave]] Functions, Jensen's inequality states the following relationship between $g(\mathbb E[X])$ and $\mathbb E[g(X)]$. - If $g$ is convex: $g(\mathbb E[X]) \le \mathbb{E}[g(X)]$. - If $g$ is concave: $g(\mathbb E[X]) \ge \mathbb{E}[g(X)]$. ![[jensens-inequality.png|center|300]] This inequality has profound implications in probability, optimization, and statistics, especially for bounding expectations and demonstrating relationships between [[Random Variable|random variables]].