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732A83 Statistical Methods
Course information
Autumn 2024
This is the course page for the course in Statistical Methods.
The first course occasion will be in S26 on Monday 2024-09-02 10:15-12 Lecture!
Course Content
The course aims to provide the student with a theoretical basis of statistical concepts and methods that are required for qualified work and research in statistics. More specifically, the course includes:
- The concept of probability
- Bayes's theorem
- The concept of a random variable, common statistical univariate and multivariate distributions and their properties
- The Multivariate normal distribution
- Functions of random variables
- Markov, Chebyshev's inequalities
- Convergence of random variables
- The concept of a characteristic function
- The Central Limit Theorem
- Point estimation: properties and methods
- The Law of Large Numbers
- The concept of consistency
- Interval estimation
- Hypothesis testing
- Simple and multiple linear regression
- Least squares estimation
- Residual and outlier analyses
- The likelihood, prior and posterior distribution
- The concept of a stochastic process
- Markov chains
- Counting processes, the Poisson process
- Continuous time stochastic processes
Course literature
Probability and Statistics for Computer Science by M. Baron (B).
Mathematical Statistics with Applications by D. Wackerly, W. Mendenhall, R. Schaeffer (WMS).
Probability and Random Processes by G. Grimmett, D. Stirzaker (GS).
Course structure
The course contains 17 lectures, and 11 exercise sessions.
All classes are planned to be on campus.
The course contains two teaching activities:
- Lecture (Fö) Introduction of concepts.
- Exercise session (LE/SE) Presentation of solutions exercises.
Lectures
The following content will be presented on each lecture:
| Lecture | Time and place | Material | |
|---|---|---|---|
| 1 | 2024-09-02 10:15 S26 | Introduction to probability, concept of probability | |
| 2 | 2024-09-05 R41 08:15 | Conditional distributions, Bayes's theorem | |
| 3 | 2024-09-10 A31 13:15 | Univariate distributions, Functions of random variables, change of variables | |
| 4 | 2024-09-13 R41 15:15 | Multivariate distributions, functions of random variables, change of variables | |
| 5 | 2024-09-17 R41 13:15 | Multivariate Normal Distribution, functions of Gaussian random variables, Conditional distribution, Simulating normal random vectors, Wishart distribution | |
| 6 | 2024-09-20 R41 15:15 | Simulating random variables with R | |
| 7 | 2024-09-23 R41 10:15 | Markov, Chebyshev's inequalities, convergence, definition of characteristic function, Central Limit Theorem | |
| 8 | 2024-09-24 R41 13:15 | Point estimation, law of large numbers, consistency | |
| 9 | 2024-10-01 R41 13:15 |
Interval estimation and hypothesis testing | |
| 10 | 2024-10-04 P42 10:15 |
Confidence intervals, normal approximation | |
| 11 | 2024-10-07 E326 10:15 | Hypothesis testing, p-value | |
| 12 | 2024-10-08 R41 13:15 | Maximum likelihood and Bayesian estimation | |
| 13 | 2024-10-14 R41 10:15 | Regression | |
| 14 | 2024-10-15 R41 13:15 | Regression | |
| 15 | 2024-11-12 R41 15:15 | Introduction to stochastic processes, examples of stochastic processes, Markov Chains | |
| 16 | 2024-11-18 U2 10:15 | Counting processes, Poisson process | |
| 17 | 2024-11-21 |
Continuous time stochastic processes |
Exercise sessions
Below is the list of exercises for each exercise session. Notice that not all exercises will be done at the session. They will be selected from this list. Emphasized exercises will be done during the session. For the bonus point sessions students aiming for these points have to be prepared with ALL the exercises for the given day.
Thursday, September 12 08:15-10:00 in S26
Type: Exercise session 1Exercise sheet: download here
Content: Exercises (WMS) 1.2, 1.22, 1.32;
2.10, 2.41, 2.42, 2.55, 2.64, 2.65, 2.68, 2.69, 2.73, 2.77, 2.79, 2.80, 2.81, 2.82, 2.83, 2.97, 2.98, 2.102, 2.104, 2.114, 2.121, 2.125, 2.126, 2.127, 2.137, 2.149, 2.163, 2.164, 2.165;
3.12, 3.15, 3.16, 3.17, 3.24, 3.33, 3.34, 3.50, 3.59, 3.61, 3.70, 3.73, 3.121, 3.122, 3.215;
4.8, 4.11, 4.12, 4.16, 4.28, 4.29, 4.30, 4.43, 4.48, 4.58, 4.69, 4.73, 4.74, 4.88, 4.94, 4.96, 4.110, 4.123, 4.126;
Thursday, September 19 08:15-10:00 in R41
Type: Exercise session 2Exercise sheet: download here
Content: Exercises (WMS) 5.4, 5.6, 5.7, 5.8, 5.9, 5.25, 5.26, 5.31, 5.34, 5.51, 5.56, 5.57, 5.75, 5.79, 5.82, 5.94, 5.100, 5.103;
Exercises (B) 4.16, 4.17, 4.18;
Thursday, September 26 08:15-10:00 in R41
Type: Exercise session 3, 2 bonus points opportunityExercise sheet: download here
Content: Exercises (WMS) 3.167, 3.169, 3.170, 3.178;
4.149, 4.150;
7.43, 7.44, 7.77, 7.78;
8.8, 8.10, 8.36;
9.3, 9.6, 9.15, 9.19, 9.20, 9.32;
Thursday, October 3 08:15-10:00 in U2
Type: Exercise session 4, 2 bonus points opportunityExercise sheet: download here
Content: Exercises (WMS) 7.26, 7.38, 7.46;
8.40, 8.44, 8.58, 8.60, 8.61;
Exercises (B) 9.9a, 9.10a, 9.12a;
Thursday, October 10 08:15-10:00 in S26
Type: Exercise session 5, 2 bonus points opportunityExercise sheet: download here
Content: Exercises (WMS) 9.74, 9.80, 9.82, 9.85, 9.90, 9.97;
10.6, 10.18, 10.19;
16.6, 16.7, 16.9;
Exercises (B) 9.1, 9.5, 9.9a, 9.10a, 9.12;
10.31, 10.32, 10.37, 10.38, 10.39, 10.40;
Thursday, October 17 08:15-10:00 in R41
Type: Exercise session 6Exercise sheet: download here
Content: Exercises (WMS) 11.3, 11.4, 11.23, 11.26, 11.30, 11.40, 11.47, 11.68, 11.76;
Thursday, October 24 08:15-10:00 in R41
Type: Exercise session 7Content: Exercises not done in the previous sessions
Thursday, November 7 08:15-10:00 in U2
Type: Exercise session 8Content: Exercises not done in the previous sessions
Thursday, November 14 08:15-10:00 in R41
Type: Exercise session 9Exercise sheet: download here
Content: Exercises (B) 6.1, 6.2, 6.3, 6.4, 6.6, 6.8;
Tuesday, November 19 13:15-15:00 in R41
Type: Exercise session 10Exercise sheet: download here
Content: Exercises (B) 6.9, 6.10, 6.11, 6.14, 6.20, 6.21, 6.23;
Monday, Novemeber 25 10:15-12:00 in R41
Type: Exercise session 11, 2 bonus points opportunityPart of exercise sheet: download here
Content: Exercises (B) 6.25;
Exercises (GS) 3 from p. 371, 18 from p. 410;
Remaining exercises:
Exercise I
Let X[1],X[2],... be an i.i.d. sequence of random varaibles satisfying P(X[i]=i)=P(X[i]=-i)=1/2. Define S[n]=X[1]+...+X[n] and T[n]=S2[n]. Is T[n] a martingale? Does there exist a sequence t[n] such that T[n]-t[n] is a martingale?
Exercise II
Let B(t) be a standard Brownian motion and F(t) be the family of sigma-algebras generated by B(s) for 0≤s≤t. Show that M(t) = B(t)2-t is a martingale with respect to F(t).
Exercise III
Consider a pure birth branching process, Z[n], with random offspring number with mean m., Show that m(-n)Z[n] is a martingale.
Exercise IV
Show that the minimum of a finite number of i.i.d. exponential random variables is exponential and find its rate.
Exercise V
Let B(t) be a standard Brownian motion and F(t) be the family of sigma-algebras generated by B(s) for 0≤s≤t. Is euB(t) a martingale?
Exercise VI
Let B(t) be a standard Brownian motion and F(t) be the family of sigma-algebras generated by B(s) for 0≤s≤t. Is cB(t/(c*c)), for c≤1 a martingale?
Active participation in the bonus point marked exercise sessions gives a maximum of the allocated bonus points per session to the exam. Active participation means that a student comes prepared to the exercise session with all the given day's exercises (please hand-in your solutions to me at the beginning of the session; or if you did not solve them on paper, e-mail them to me before the session), correctly solves an exercise on the board, is able to answer questions about the presented solution and is able to give help and comments to the classmates' presented solutions. In the sessions, for each exercise a student will be selected (how depends on the number of students) to present a solution.
Physical presence at the excersise sessions is a necessary condition to obtain the bonus points.
The same system will be used for the Advanced R Programming course Computational Complexity exercise session and the Bioinformatics course's exercise sessions.Written exam
The exam will be a written exam on 2024-12-05
The examination has max score 20 points and grade limits: A : 18p, B: 16p, C: 14p, D: 12p, E: 10p.The material that will be included with the exam can be downloaded here.
Previous exams can be found here.
Help materials for written exam: On the exam you will be allowed to have one double-sided A4 page of own handwritten (i.e., NOT printed out) notes.
Eight bonus points can be obtained in total from the exercise sessions (bonus point sessions marked).
Reporting on the exam a probability outside the [0,1] interval or a negative variance without a comment that a mistake must have been made somewhere will result in a NEGATIVE point for each such instance.
Staff
- Krzysztof Bartoszek, examiner
- Krzysztof Bartoszek, lecturer
- Johan Alenöv, lecturer
- Bayu Beta Brahmantio, teaching assistant
Page responsible: Krzysztof Bartoszek
Last updated: 2024-07-12
