Introduction to Probability Theory

Undergraduate course, IUT Orsay, 2020

An introduction to probability theory concluding with an introduction to statistical tests.

Foundations of Probability

  • Probability Axioms
  • Events
  • Mutually Exclusive Events
  • Law of Total Probability
  • Conditional Probability
  • Bayes’ Rule
  • Independent Events

Discrete Random Variables

  • Definition of a Random Variable
  • Cumulative Distribution Functions
  • Examples: Discrete Uniform, Bernoulli, Binomial, Geometric
  • Independent Random variables

Jointly Distributed Random Variables

  • Marginal distributions
  • Conditional distributions
  • Sum and product of Random Variables
  • Expectation and variance of Random Variables
  • Correlation

Continuous Random Variables

  • Probability Density Functions
  • Examples: Continuous Uniform, Gaussian
  • Function of random variables
  • Change of variables

Convergence

  • Convergence in law, Convergence in probability
  • Central limit theorem, Weak law of large numbers
  • Poisson distribution
  • Approximation of Binomial distribution by Poisson or Gaussian distributions

Statistical tests

  • Principles of statistical tests
  • Decision rule, risk, power, p-value
  • Test comparisons