Empirical Methods and Statistics

Course materials, lecture notes, exercises and solutions for the Empirical Methods and Statistics course at Sofia University — Software Engineering Faculty. Designed for beginners who want to build a solid foundation in probability theory and statistics.

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Table of Contents

• Lectures
• Exercises
• Exams
• Homeworks
• Additional Problems
• Additional Material
• Interview Tasks
• References

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Lectures

#	Date	PDF
01	2020-10-01	SEM L01
02	2020-10-08	SEM L02
03	2020-10-15	SEM L03
04	2020-10-22	SEM L04
05	2020-10-29	SEM L05
06	2020-11-05	SEM L06
07	2020-11-12	SEM L07
08	2020-11-19	SEM L08
09	2020-11-26	SEM L09
10	2020-12-03	SEM L10
11	2020-12-10	SEM L11
12	2020-12-17	SEM L12
13	2021-01-07	SEM L13
14	2021-01-14	SEM L14
15	2021-01-21	SEM L15

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Exercises

Each week has two tracks: Statistics & Empirical Methods practical exercises and Statistical Computing with R.

Week	Statistics & Empirical Methods	Statistical Computing with R
01	1.1. Combinatorics – Problems 1.2. Combinatorics – Solutions 1-9 1.3. Combinatorics – Solutions 10	1.0. Introduction 1.1. Basic Syntax 1.2. Data Types 1.3. Data Structures 1.4. Missing Data 1.5. Packages 1.6. Reading Data 1.7. Descriptive Statistics 1.8. Examples 1.9. Verzani Problem Set 1.10. Vectors (Moodle Tasks) 1.11. Vectors (Moodle Tasks Solutions)
02	2.1. Combinatorics part 2 – Problems 2.2. Combinatorics part 2 – Solutions 1-7 2.3. Probabilities – Problems	2.1. Univariate Data 2.2. Verzani Problem Set 2.3. Moodle Tasks 2.4. Moodle Tasks Solutions
03	3.1. Conditional Probabilities, Independent Events – Problems 3.2. Conditional Probabilities, Independent Events – Solutions 3.3. Exercise 3 – Problems 3.4. Exercise 3 – Solutions 1-7	3.1. Bivariate Data 3.2. Verzani Problem Set 3.3. Moodle Tasks 3.4. Moodle Tasks Solutions
04	4.1. Conditional Probabilities bis – Problems 4.2. Conditional Probabilities – Solutions 5-8 4.3. FMI – PTMS 1-2-3 Solutions 4.4. FMI-PTMS 4 – Problems 4.5. FMI-PTMS 4 – Hints and Solutions 4.6. FMI-PTMS 4 – Solutions	4.1. Multivariate Data 4.2. Moodle Tasks
05	5.1. Bayes Law, Geometric Probability – Problems 5.2. Bayes Law, Geometric Probability – Solutions 1-7 5.3. Bayes' Law, Geometric Probability – Solutions 8-9 5.4. Bayes Law, Geometric Probability – Hints 5.5. FMI – PTMS 5 – Problems 5.6. FMI – PTMS 5 – Solutions 5.7. FMI – PTMS 5 – Detailed Solutions 5.8. Solutions with Drawings	—
06	6.1. Geometric Probability. Discrete Random Variables 6.2. Discrete Random Variables – Problems 6.3. Discrete Random Variables – Solutions	6.1. Random Data (Variables) 6.2. Verzani Problem Set 6.3. Moodle – Discrete Random Variables – Problems 6.4. Moodle – Discrete Random Variables – Solutions 6.5. Moodle – Continuous Random Variables – Problems 6.6. Moodle – Continuous Random Variables – Solutions
07	7.1. Discrete Random Variables part 2 7.2. Discrete Random Variables – Problems 7.3. Discrete Random Variables – Solutions 7.4. First Control Test Preparation (Past Problems and Solutions)	—
08	8.1. Discrete Random Variables part 3 8.2. FMI PTMS 8 – Problems 8.3. FMI PTMS 8 – Solutions	8.1. Limit Theorems. Convergence. Normal Distribution Tests
09	9.1. Discrete Random Variables part 4 9.2. FMI – PTMS 9 – Problems 9.3. FMI – PTMS 9 – Solutions 9.4. Control Test 1, variant 1, groups 1-3 – Problems 9.5. Control Test 1, variant 1 – Detailed Solutions 9.6. Control Test 1, variant 2, groups 1-3 – Problems 9.7. Control Test 1, groups 4-5 – Problems 9.8. Control Test 1, groups 4-5 – Solutions	9.1. Confidence Interval Estimation 9.2. Verzani Problem Set 9.3. Moodle – Confidence Intervals – Problems 9.4. Moodle – Confidence Intervals – Solutions
10	10.1. Continuous Random Variables, groups 4-5, part 1 10.2. FMI – PTMS 10 – Problems 10.3. FMI – PTMS 10 – Solutions	10.1. Hypothesis Testing 10.2. Verzani Problem Set
11	11.1. Continuous Random Variables, groups 4-5, part 2 11.2. FMI PTMS 11 – Problems 11.3. FMI PTMS 11 – Solutions	11.1. Two-sample Hypothesis Testing 11.2. Verzani Problem Set 11.3. Moodle – Hypothesis Testing (two samples) – Problems 11.4. Moodle – Hypothesis Testing (two samples) – Solutions
12	12.1. Continuous Random Variables part 3	12.1. Chi-square Tests 12.2. Verzani Problem Set
13	13.1. Continuous Random Variables part 4 13.2. Continuous Random Variables, FMI – PTMS 12-14	13.1. Regression Analysis 13.2. Verzani Problem Set
14	14.1. Continuous Random Variables part 5 14.2. FMI PTMS 12-14 – Problems 14.3. FMI PTMS 12-14 – Solutions	14.1. Multiple Linear Regression 14.2. Verzani Problem Set
15	15.1. Control Test 2 Preparation 15.2. Control Test 2, groups 1-3 – Problems 15.3. Control Test 2, groups 1-3 – Detailed Solutions 15.4. Control Test 2, groups 4-5 – Problems 15.5. Control Test 2, groups 4-5 – Solutions	15.1. Analysis of Variance (ANOVA) 15.2. Verzani Problem Set

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Exams and Preparation for Exams

Date	Materials
—	Exam Preparation – Problems Exam Preparation – Solutions
2021-01-27	SEM Final Exam – Problems and Solutions
2021-02-03	SEM Final Exam – Problems SEM Final Exam – Solutions Detailed Solutions for Problems 4, 6, 7, 9

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Homeworks

Groups	Materials
1–3	SEM HW groups 1-3 – Problems SEM HW groups 1-3 – Solutions
4–5	SEM HW groups 4-5 – Problems

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Additional Problems

#	Problem
01	Crux Mathematicorum Vol. 42, No. 6, June 2017, p. 11 (Geometric Probability)
02	Crux Mathematicorum Vol. 47, No. 1, January 2021, pp. 13-18 (Games, Geometric Probability)
03	Number of Sixes Probability (Chebyshev & CLT)
04	Fair Die Distribution
05	2X–3Y
06	Coprime Natural Numbers Probability
07	Deck of Cards Questions
08	Basic Randomization
09	x/y Closer to Even Integer Probability (Putnam 1993 B3)
10	Dice Process
11	Probability of Winning at Craps

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Additional Material

#	Title
01	Central Limit Theorem
02	Berry-Esseen Bounds — Analysing the Speed of Convergence of CLT
03	Radon-Nikodym Theorem

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Interview Tasks

Problems	Solutions
Interview Problems (Game-Math Designer, Quant Developer, Mathematician)	Solutions
Line Calc (iGaming)	Program
Expected Bonus Win (iGaming)	—
Musicians & Concerts Problem	Musicians & Concerts Solution

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References

#	Source
[1]	Empirical Methods and Statistics lecture notes @ FMI — Sofia University, Software Engineering. Lecturer: Mladen Savov
[2]	Probabilities Theory — Exercises, Emil Kamenov, Miroslav Stoenchev
[3]	Monika Peteva Petkova's notes on R programming language
[4]	Martin Minchev's notes on Probabilities and Statistics
[5]	SimpleR — Using R for Introductory Statistics, John Verzani
[6]	Crux Mathematicorum, Canadian Mathematical Forum
[7]	Probability and Random Processes, Geoffrey R. Grimmett & David R. Stirzaker, 3rd Edition
[8]	One Thousand Exercises in Probability, Geoffrey R. Grimmett & David R. Stirzaker
[9]	Dobromir Pavlov Kralchev's notes on Combinatorics and Generating Functions
[10]	William Lowell Putnam Mathematical Competition
[11]	102 Combinatorial Problems, Titu Andreescu & Zuming Feng
[12]	Personal notes, solutions and opinionated approaches for problem solving