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