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Jensen's Inequality

Drag control points to reshape a convex or concave function, place probability mass, then compare φ(𝔼[X]) to 𝔼[φ(X)] directly.

jensen's inequality
convexity
expectation
interactive proof

Neither

Challenges

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Convex φ — drag the ◆ control points until the curve is convex (U-shaped). Jensen's inequality then guarantees φ(𝔼[X]) ≤ 𝔼[φ(X)] for any placement of X.

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Concave φ — make the curve concave (∩-shaped). Jensen's inequality reverses: φ(𝔼[X]) ≥ 𝔼[φ(X)].

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Both directions — find a curve that is neither convex nor concave. Then, with that same curve, find two different placements of X: one where φ(𝔼[X]) ≤ 𝔼[φ(X)] and one where the inequality flips. Click Show Jensen's for each placement.

Drag ◆ to reshape φ · Click the x-axis to place probability mass · Right-click axis to remove a point