Introduction to probability theory

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 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