6PAS1 Probabilty and Statistics 1, winter semester 2025/2026
A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours, unless specified otherwise)
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 7 November 18, 2025
Expectation
and variance of binomial distribution.
Geometric distribution,
exponential distribution
(definition, computation of expectation and variance for each distribution).
Memoryless property
of geometric 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.
6PAS1 Probabilty and Statistics 1, winter semester 2024/2025
Topics not covered the last semester
Direct product of sample spaces.
Datasets and their graphical representations.
Sample mean, sample variance, sample correlation (unbiased estimators).
Winter semesters from 2017/2018 till 2019/2020:
Created: Sun Sep 21 2025
Last modified: Wed Oct 14 2026 19:37:20 CEST