Concentration Inequalities - Interactive Visualization

Concentration inequalities bound the probability that a random variable deviates far from its mean. Markov's inequality requires only E[X]≥0. Chebyshev requires finite variance. Hoeffding applies to bounded variables and improves exponentially with sample size. These bounds are the mathematical foundation of PAC learning theory - they guarantee that a model trained on n samples will generalize.

• Compare Markov, Chebyshev, and Hoeffding bounds on the same plot
• See how each bound becomes tighter or looser vs true tail probability
• Adjust σ to see variance effect on Chebyshev bound
• Adjust n to see Hoeffding's exponential improvement with sample size
• Foundation for PAC learning, generalization bounds, and statistical guarantees

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