Probability I

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Every fall semester

Develops the probability basis requisite in modern statistical theories and stochastic processes. Topics of this course include measure and integration foundations of probability, distribution functions, convergence concepts, laws of large numbers and central limit theory. (1st of two courses)


MATH 6337 or equivalent

Course Text: 

At the level of Billingsley:  Probability and Measure.

Topic Outline: 
  • Classes of sets, including sigma fields and the Dynkin system theorem
  • Probability measures, including basic properties, generation of probability measures, and connections to distribution functions
  • Random variables, random vectors and discrete-parameter stochastic processes, including basic properties and connections to probability measures and distribution functions
  • Expectation, including basic properties, convergence theorems and inequalities
  • Independent random variables, including basic properties, connections to infinite-dimensional product measures and Fubini's theorem
  • Modes of convergence of random variables, including almost everywhere convergence, the Borel-Cantelli Lemma, convergence in probability, and convergence in L^p
  • Laws of large numbers, including weak and strong laws of large numbers, Kolmogorov's inequality, equivalent sequences and random series
  • Convergence in distribution, including basic properties, and connections to sequential compactness, tightness, and uniform integrability
  • Characteristic functions, including basic properties and connections to probability measures
  • Central Limit Theorems, including use of characteristic functions and Lindeberg's theorem