Course content and schedule

Part I, probability theory.

Lecture	Date	Content	Material
1	3/9	Introductory lecture.	Ch 1.
2	5/9	Set theory. Union, intersection, complements of sets. Elementary definition of probabilities. Why the elementary definition is not satisfactory. Strict definition of probabilities: Sigma-algebras	2.1, 2.2.
3	10/9	Strict definition of probabilities: Sigma-algebras, probability measures. Properties of probability measures	2.3
4	12/9	Conditional probabilities. Independent events.	3.1-3.2
5	17/9	Measurability. The distribution of a r.v. Properties of the distribution function.	4.1-4.2
6	19/9,	Properties of the distribution function. Counting and length measure, and integrals with such. The density of a r.v.	4.2-4.4
7	20/9	Random vectors. The density of a random vector. Simultaneous, and marginal distributions.	5.1-5.2
8	26/9	Independent random variables. Conditional distributions.	5.3-5.4
9	27/9	Functions of r.v. and random vectors. Maxima, minima. Sums.	6.1-6.2
9.5	3/10, 10:15-12:00	Maxima, minima. Sums. The Riemann-Stieltjes integral	6.2, 7.1
10	3/10, 13:15-15:00	The Riemann-Stieltjes integral. Expectation.	7.1-7.2
11	4/10	The multivariate Riemann-Stieltjes integral. Expectations and covariances.	7.3-7.4 (7.5)
12	9/10	Conditional expectation. Examples of distributions.	8, 9.1-9-5
13	10/10	Examples of distributions. Notions of convergence.	9.6-9.10,10
14	17/10	Convergence in probability and in distribution. The law of large numbers. The central limit theorem.	10

Exercise	Date	Exercise number
1	6/9	Chapter 2: Exercise 1-8
2	13/9	Ch 3: Ex 1-3, 5-6
3	24/9	Ch 4: Ex 1,2,3,4,5,7,
4	1/10	Ch 4: 6, Ch 5: 1-5, Extra exercises on Chapter 2 from this list-ex-ch2: 12
5	8/10	Ch 6: 1,2,3,4, Extra exercises on Chapter 2 from the list above: 1,5,6,7,8
6	15/10	Ch 7: 1,2,5,7,9,11,14,15
7	22/10, 10:15-12:00	Ch 8: 1,2,3,5, Ch 9: 1,3,5, Ch 10: 1,2.
8	22/10, 13:15-15:00	Exams Probability theory, 171024, 171113

Lecture	Date	Content	Material
1		Introductory Lecture. Statistics. Properties of statistics. Unbiasedness. Consistency.	Ch 13.1-13.3
2		Properties of statistics. Plug-in estimators. The standard error.	13.3, 13.4
3		Least squares and maximum likelihood estimators.	14
4		Confidence Intervals	15.1
5		Joint confidence Intervals. Tests.	15.2, 16
6		Tests. The power function. Composite null hypotheses.	16.1-16.2
7		p-values. Repeated tests. Normal approximations of estimators.	16.3-16.5, 17
8		Applications to some common situations; Gaussian data, binomial data, Poisson data.	18
9		Test-based confidence intervals and the confidence interval method.	19
10		Parametric, semi- and non-parametric problems. The empirical distribution function.	20, 21, R code illustrating properties of the empirical distribution function, and Donskers result can be found here.
11		Some nonparametric inference problems; kernel estimators and k-sample tests.	22.1-22.4
12		Nonparametric problems	22.1-22.4
13		k-sample tests, Kaplan-Meier estimator	22.5
14	Mon, 17/12, 10:15-12.00	Linear regression.	23.1-23.3, 7.5
15	Tue, 18/12, 10:15-12:00	Linear regression.	23.4-23.6
16	Tue, 18/12, 13:15-15:00	Repetition