Probability and Statistics – Introductory Lecture Vocabulary

Course Administration

• Department: Microsystems Engineering (IMTEK), University of Freiburg
• Course: Probability & Statistics (Code 11LE50V-6100 – Lecture, 11LE50Ü-6100 – Exercise)
• Lecturer: PD Dr. Cristian Pasluosta
  • Laboratory for Biomedical Microtechnology
  • Office: Room 201\text{-}00\text{-}022 / Phone: 7251
  • Email: cristian.pasluosta@imtek.de
• Tutor / Exercise leader: Thomas Krauskopf
  • Email: thomas.krauskopf@imtek.uni-freiburg.de
• Contact & consultation hours: by appointment (via the above e-mails)

Assessment & Participation

• Written final exam
  • Weight: 100\% of final mark
• Admission requirement:
  • Must achieve \ge 50\% of the points offered in the weekly exercises
• No partial oral exam or bonus scheme was announced

Weekly Calendar (WS 2024/2025)

• Lecture: Tuesdays 12:00 – 14:00
• Exercise: Fridays 12:00 – 14:00 (begin 25 Oct 2024)

Learning Material Distribution

• Lecture videos uploaded to Ilias (password: anova)
• Hard-copy or PDF literature placed on the portal and/or in the library

Recommended Literature ("Dennis’ Choice")

• Norbert Henze — Stochastik für Einsteiger (12th ed., Springer, 2018)
  • Cover formula shown: B(X)=\sqrt{x}\,f(x)\,dx
• Lothar Papula — Mathematik für Ingenieure & Naturwissenschaftler 3
• Herold Dehling & Beate Haupt — Einführung in die Wahrscheinlichkeitstheorie und Statistik
• Jürgen Bortz & Christof Schuster — Statistik für Human- und Sozialwissenschaftler
• W. J. Conover — Practical Nonparametric Statistics (3rd ed.)
• Miller & Freund / R. A. Johnson — Probability & Statistics for Engineers (9th ed.)
• Jim Pitman — Probability (Springer Texts in Statistics)
• Aeneas Rooch — Statistik für Ingenieure

Software Helpers

• Open-source: R, Gretl, SageMath, SymPy, Maxima, Axiom
• Commercial / proprietary: Matlab (+ Statistics Toolbox), Mathematica, SPSS, Origin, Prism, MiniTab
• Mobile (iPad): StatsMate, StatViz

Web Resources

• http://www.statsoft.com/textbook/
• http://stattrek.com/
• http://www.r-project.org/
• https://www.graphpad.com/data-analysis-resource-center/
• https://www.real-statistics.com
• https://sites.psu.edu/shinyapps/

Lecture 1: Intuition, Events & Probabilities

Everyday Motivation

• Games of chance (lottery, roulette, poker, black jack)
• Weather forecasts (e.g., “70\% chance of rain”)
• Stock markets & insurance premiums
• Political stalemates resolved by drawing lots (example: state parliament of Rhineland-Palatinate)
• Engineering reliability: Mean-Time-Between-Failures (MTBF)
• Scientific phenomena:
  • Diffusion of an ink drop vs. molecular motion
  • Radioactive decay: single \mathrm{^{14}C} atom vs. macroscopic sample

Recent & High-impact Applications

• Medicine: reported therapy success rates at 95\% confidence
• Biology: experimental design & statistical analysis
• Traffic engineering: queuing theory for jams
• Epidemiology: disease-spread modelling
• Physics mega-projects (CERN, Manhattan, Apollo)

Historical Milestones

• Ancient Greeks & Romans used bone dice
• 1657 – Christiaan Huygens: Tractatus de Ratiociniis in Ludo Alea
• 1749{-}1827 – Pierre-Simon Laplace: classical (Laplace) definition of probability
• 1903{-}1987 – Andrey N. Kolmogorov: axiomatic foundation (three axioms)
• 1994 Nobel Prize in Economics: Nash / Selten / Harsanyi (Game Theory)

What is Stochastic?

• Deals with events that cannot be predicted deterministically due to incomplete information
• Two branches:
  • Probability theory: modelling likelihood of outcomes
  • Statistics: analysing data arising from experiments

Ideal Random Experiment (Fisher, 1935)

• Conducted under well-defined, reproducible conditions
• Complete set of possible outcomes is known beforehand
• In principle repeatable infinitely often under identical conditions

Fundamentals of Set Theory

Basic Terminology

• Sample space: \Omega — the set of all possible outcomes
• Elementary outcome: \omega (element of \Omega)
• Subset: A\subseteq B means every \omega\in A is also in B
• Superset: B\supseteq A (inverse relation)
• Power set \mathcal P(S): set of all subsets of S (includes \varnothing and S itself)
• Cardinality |A|: number of elements in a finite set A

Example

• If S={a,b,c} then \mathcal P(S)={\varnothing,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}
  • |S|=3, |\mathcal P(S)|=2^3=8

Partition of a Set

• A collection {A1,A2,\dots,An}\subseteq\Omega is a partition of A\subseteq\Omega if \bigcup{i=1}^{n} Ai = A Ai\cap A_j = \varnothing\quad (i\neq j)

Common Operators

• Union: A\cup B — \omega\in A\text{ or }\omega\in B (or both)
• Intersection: A\cap B — \omega in both A and B
• Complement: A^c=\Omega\setminus A
• Set difference: B\setminus A (elements in B not in A)
• Empty set: \varnothing (impossible event)
• Whole space \Omega (sure event)

Venn-Diagram Conventions

• John Venn (1834–1923) visualised relations using overlapping circles

Algebraic Set Identities

• Commutative:
  A\cap B = B\cap A , A\cup B = B\cup A
• Associative:
  A\cap(B\cap C)=(A\cap B)\cap C , A\cup(B\cup C)=(A\cup B)\cup C
• Distributive:
  A\cap(B\cup C)=(A\cap B)\cup(A\cap C)
• De Morgan:
  (A\cup B)^c = A^c\cap B^c , (A\cap B)^c = A^c\cup B^c
• Disjointness: A\cap B=\varnothing ⇒ events cannot co-occur

Probability-Related Notions (Preview of Upcoming Lectures)

• Probability measure P, probability mass/function p(\omega)
• Kolmogorov axioms (1–3): non-negativity, normalisation P(\Omega)=1, countable additivity
• Laplace probability: P(A)=\dfrac{|A|}{|\Omega|} when outcomes are equally likely
• Odds & ratio interpretation (\emph{"ratio-checker"} tool)
• Basic distributions arising from Laplace principle:
  • Discrete uniform
  • Continuous (Laplace) distribution
• Combinatorial building blocks (to be covered):
  • Ordered / unordered sampling
  • With / without replacement

Ethical & Practical Considerations

• Statistics underpins medical decisions (therapy success at 95\% confidence)
• Misinterpretation of probability can lead to policy errors (election polling, insurance risk)
• Rigorous experimental design (Fisher) is essential for reproducibility

Connections & Future Outlook

• Set theory & logic → foundation for Kolmogorov axioms
• Combinatorics → counting techniques for probability calculation
• Limit theorems (LLN, CLT) → bridge between probability & statistics
• Estimation (MLE, MoM) → will rely on probability models developed here