Jensens Inequality

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This graph illustrates Jensen's Inequality, a fundamental concept in mathematical optimization and probability theory. It compares a concave function (blue curve, representing the natural logarithm function) with a linear function (orange dashed line). The inequality states that for a concave function, the function of an expectation is always greater than or equal to the expectation of the function, i.e. $f(E[X]) \geq E[f(X)]$, depicted here by the fact that the dashed line (secant) is always below the curve. Equality holds if and only if the random variable is a constant (i.e. there is no randomness).