# Introduction to Probability Theory

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