# 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
- Bayesâ€™ 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