6PAS1 Probabilty and Statistics 1, winter semester 2024/2025
A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours if not specified otherwise; tutorials
were running separately)
Class 3 October 7, 2024
Direct product of sample spaces. Examples of calculation of probabilities.
Class 5 October 21, 2024
Notion of discrete random variable, mass function, and distribution function.
Examples. Properties of discrete distribution functions: piecewise constant,
non-decreasing, \(\lim_{x\to -\infty} F_X(x) = 0\),
\(\lim_{x\to +\infty} F_X(x) = 1\).
Class 6 November 4, 2024
Continuous random variable. Interplay between discrete and continuous.
Continuous distribution function, density function, its properties. Density is
not probability!
Definition of expectation, variance, and standard deviation of discrete and
continuous random variable.
Class 7 November 11, 2024
Alrernative formula for variance: \(Var(X) = E[X^2] - E[X]^2\).
Change of variables formulas for expectation and variance. Linear change of
variables.
Discrete unform distribution, its expectation and variance.
Class 8 November 18, 2024
Formulas for sums
of powers.
Continuous uniform distribution, binomal distribution, geometric distribution,
exponential distribution (definition, computation of expectation and variance
for each distribution).
Class 9 November 25, 2024
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 10 December 2, 2024
Joint mass function, joint distribution function. Independent random
variables. If \(X\) and \(Y\) are indepenent random variables, then
\(E[XY] = E[X]E[Y]\). Covariance and correlation.
Class 11 December 9, 2024
Properties of correlation. Independent random variables are uncorrelated, but
not vice versa.
Datasets and their graphical representations.
Sample mean, sample variance, sample correlation (unbiased estimators).
Correlation is not causation.
Class 12 December 16, 2024
Statistical model, linear statistical model (linear regression), method of least
squares. Hypothesis testing, test statistic, null and alternative hypothesis,
\(p\)-value, type I and type II error.
Winter semesters from 2017/2018 till 2019/2020:
Created: Sun Sep 21 2025
Last modified: Tue Oct 14 2025 17:58:39 CEST