# 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
- 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