1.      Events and Probability

1.1  Revision of basic concepts (sample space, algebra of events, probability measure, permutations and combinations, sampling with or without replacement, conditional probability, partitions of the sample space, theorem of total probability, Bayes' Theorem, independence)

1.2  Kolmogorov's Axioms (infinite sample space)

 

2.      Random Variables and Distribution Functions

2.1  Random variables. Distribution function

2.2  Univariate distributions - discrete, continuous, mixed.

2.3  Transformations of a single random variable

2.4  Joint distributions - discrete and continuous

2.5  Marginal and conditional distributions. Independence.

2.6  Transformations of jointly distributed random variables

 

3.      Expectation

3.1  Expectation. Properties of expectation

3.2  Moments and parameters

3.3  Markov's and Chebyshev's inequalities

3.4  Moment generating function

3.5  Characteristic function

3.6  Conditional expectation

 

4.      Some Discrete and Continuous Distributions

4.1      The Uniform Distribution (discrete)

4.2      The Bernoulli Distribution

4.3      The Binomial Distribution

4.4      The Geometric Distribution

4.5      The Negative binomial Distribution

4.6      The Hypergeometric Distribution

4.7      The Multinomial Distribution

4.8      The Poisson Distribution

4.9      The Uniform Distribution (continuous)

4.10  The Normal Distribution

4.11  The Bivariate Normal Distribution

4.12  The Multivariate Normal Distribution

4.13  The Log-normal Distribution

4.14  The Exponential Distribution

4.15  The Gamma Distribution

4.16  The Chi-Squared Distribution

4.17  The Erlang Distribution

4.18  Student's t-Distribution

4.19  The F-Distribution

  

5.      Sequences of Random Variables and their Convergence

5.1  Sequences of random variables

5.2  Modes of convergence

5.3  Bernoulli's Theorem. Laws of Large Numbers

5.4  Central Limit Theorem

5.5  De Moivre-Laplace theorem and the Normal approximation to the Poisson distribution