All the books are available in electronic form in multiple places.
= available in
the university library
A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours, unless specified otherwise;
tutorials are running separately)
Class 1 September 23, 2025
Organizational issues. The subject of probability and statistics. A brief
refresher of combinatorics (permutations, variations without and with
repetitions, combinations, binomial coefficients).
Class 2 September 30, 2025
Inclusion-exclusion principle. Probabilistic terminology: sample space, event,
elementary event, examples. Definition of probability, examples.
Inclusion-exclusion principle for probabilities.
Class 3 October 7, 2025
Relationship between different inclusion-exlcusion principles. Conditional
probability. Bayes' formula and its generalization. Two independent events
(equivalent definitions).
Class 4 October 14, 2025
Properties of independence of two events: behavior with respect to complement,
absence of transitivity. Independence of a set of events (equivalence of two
definitions).
Class 5 October 21, 2025
Discrete random variable, mass function, and distribution function. Examples.
Properties of discrete distribution functions: piecewise constant,
non-decreasing, \(\lim_{x\to -\infty} F_X(x) = 0\),
\(\lim_{x\to +\infty} F_X(x) = 1\).
Continuous random variable. Interplay between discrete and continuous.
Continuous distribution function, density function, its properties. Density is
not probability!
Class 6 November 11, 2025 (3 hours)
Definition of expectation, variance, and standard deviation of discrete and
continuous random variable. Alternative formula for variance:
\(Var(X) = E(X^2) - E(X)^2\).
Change of variables formulas for expectation. Linear change of variables for
expectation and variance.
Formulas for sums
of powers.
Discrete and continuous uniform distribution, its expectation and variance.
Binomial distribution.
Class 8 November 25, 2025
Memorylessness of exponential distribution. Poisson distribution. Normal
distribution, it's significance, Central Limit Theorem. Idea of a proof of the
Central Limit Theorem using characteristic function of a random variable and its
properties (mainly according to Wikipedia:
Central limit theorem,
Characteristic function).
Class 9 December 2, 2025
Joint mass function, joint distribution function. Marginal probabilities. Change
of variable formulas for \(E(g(X,Y))\). An alternative derivation of the formula
for expectation of binomial distribution using linearity of expectation.
Independent random variables.
Class 10 December 9, 2025
If \(X\) and \(Y\) are indepenent random variables, then \(E(XY) = E(X)E(Y)\).
Covariance and correlation. Properties of correlation. Correlation matrix.
Independent random variables are uncorrelated, but not vice versa. Correlation
is not causation. Method of least squares.
Class 11 December 16, 2025
Statistical model, linear statistical model (linear regression). Hypothesis
testing, test statistic, null and alternative hypothesis,
\(p\)-value, type I and type II error.
