Probability Inequality Examples

Probability · Axiom Academy

EXAMPLE Probability Inequality Examples Applying Markov, Chebyshev, and Chernoff bounds to bound tail probabilities Excellent work! You've mastered probability inequalities. Here's what we learned: Markov's Inequality: For non-negative X, P(X ≥ a) ≤ E[X]/a. Simple but weak, only requires the mean. Best for quick rough bounds. Chebyshev's Inequality: P(|X - μ| ≥ k) ≤ σ²/k². Stronger than Markov, uses variance information. Works for any distribution with finite variance. Chernoff Bound: P(X ≥ a) ≤ e^(-ta) M(t) for any t > 0, then optimize over t. Exponentially tight, requires MGF. Best bound but needs more information. Trade-offs: Markov is weakest but easiest. Chebyshev is moderate and widely applicable. Chernoff is strongest but requires MGF and optimization. Choose the right inequality based on what information you have and how tight a bound you need. Chebyshev is often the sweet spot for practical applications!

This is the written version of the interactive lesson above. See the full Probability course.