15-259/659 Probability and Computing (PnC), SPRING 2022, 12 Units

Probability theory has become indispensable in computer science.

• It is at the core of machine learning, where one often needs to make decisions under stochastic uncertainty.
• It is also integral to computer science theory, where probabilistic analysis and ideas based on randomization appear in many algorithms.
• It is a central part of performance modeling in computer networks and systems, where probability is used to predict delays, schedule resources, and provision capacity.

This course gives an introduction to probability as it is used in computer science theory and practice, drawing on applications and current research developments as motivation and context.

• Probability on Events.
• Discrete and Continuous Random Variables.
• Conditioning and Bayes.
• Higher Moments.
• Laplace transforms and z-transforms.

• Gaussians and Central Limit Theorem.
• Tails and Stochastic dominance.
• Simulation of random variables.
• Heavy-tailed distributions.

• Concentration inequalities: Markov, Chebyshev, Chernoff Bounds.
• Las Vegas and Monte Carlo randomized algorithms.

• Randomized Sorting, Min-cuts, Balls-and-bins.
• Randomized Matrix-multiplication checking.
• Randomized Primality testing, Hashing, Tournament ranking, etc.

• Discrete-time Markov Chains
• Ergodicity (with proofs)
• Continuous-time Markov chains.
• Poisson process.

• Elementary queueing theory with applications to modeling server farms, buffer provisioning, and capacity provisioning for data centers.
• Inspection paradox. PASTA.