Conditional density,
Conditional Expectation.
Bayesian Statistical Theory and Learning

The Rule of Double Expectation Conditional variance, The Formula Var(Y) = E (Var(Y|X)) + Var( E(Y | X)) and its applications.  Conditional expectation w.r.t. a sigma-field.
Bias-Variance Decomposition of data science

Characteristic function, examples and properties, table of formulas, characteristic function of a normal distribution.
Sums of a random number of random variables

Probability generating function, examples and properties

Concepts of convergence in probability theory,  
Convergence of sums and functions of random variables. Borel Cantelli, strong law of large numbers. Handout  

Multivariate Gaussian variables

Gaussian process, covariance kernel,
 the Wiener process

Kernels, Gaussian Processes in Learning

Distribution functions.
 Multivariate random variables

Multivariate random variables.
 Marginal density, Independence,
Density of a transformed random vector,
Exponential families of distributions

Some Theorems of Probability calculus.
Probability on propositional logic.  
PAC-theory of machine learning 

Sigma-fields,
Probability space.
Axioms of probability calculus

Maximum likelihood. Model choice. Confidence intervals.
Confidence intervals for model choice in machine learning 

More on Expected Risk Minimisation. Expectation – Maximisation Algorithm

Data science and high dimensional data