1. Outcomes, events, probabilities.

1.1 Sample spaces, events, axioms of probability.

1.2 Conditional probabilities, independence of events.

2. Random variables.

2.1 The concept of random variables, distribution.

2.2 Discrete random variables, standard distributions.

2.3 Continuous random variables, standard densities.

2.4 Distribution of functions of random variables.

3. Joint distributions.

3.1 Discrete joint distributions, independence.

3.2 Expectation for discrete random variables.

3.3 Continuous joint distributions.

3.4 Transformations of random variables.

4. Expectation.

4.1 Definition of expectations.

4.2 Variance and moments.

4.3 Covariance and its properties.

4.4 Conditional distributions and conditional expectation.

5. Transformations of distributions.

5.1 Generating functions.

5.2 Applications, branching processes, Panjer recursion.

6. Limit theorems.

6.1 Weak law of large numbers.

6.2 Central limit theorem.