Chapter 10: Probability Inequalities

Learning Objectives

• State and prove Markov's inequality and recognize it as the root of all tail bounds
• Derive Chebyshev's inequality from Markov and understand its limitations
• Apply the Chernoff bound via MGF optimization and compute it for Gaussian, Poisson, and binomial distributions
• State and apply Hoeffding's inequality for sums of bounded independent random variables
• Prove Jensen's inequality for convex and concave functions and apply it to information-theoretic and fading channel problems
• Compare the tightness of different probability inequalities and select the appropriate bound for a given problem