Syllabus Probability

Applied Computer Science and Artificial Intelligence; A.A. 2023/24, Lorenzo Bertini and Vittoria Silvestri

• Introduction to probability theory. Axioms and correspondence rules.
• Discrete probability spaces, events and their operations.
• Combinatorial calculus: arrangements, combinations, with and without repetition. Binomial and multinomial coefficients.
• Inclusion/exclusion and application to the number of fixed points of a random permutation.
• Product probability spaces. Independent events. Bernoulli schemes.
• Conditional probability. Formulae of composite probabilities, total probabilities, and Bayes.
• Discrete random variables: distribution, expectation, variance, and covariance.
• Independent and uncorrelated random variables.
• Bernoulli, binomial, and hypergeometric random variables.
• Sum of independent random variables.
• Geometric random variable and its loss of memory.
• Poisson variable as limit of binomials.
• Chebyshev inequality and law of large numbers.
• Joint, marginal, and conditional distributions.
• Continuous random variables: probability density, distribution function, expectation, and variance.
• Uniform random variable.
• Representation of an arbitrary random variable in terms of a uniform random variable.
• Gaussian random variables. Use of the tables for the numerical calculation of probabilities.
• Central limit theorem (without proof) and applications.

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