Mathematics BSc
Course description
2013.
Course description
2013.
Probability Theory 1
| Hours lect+pc |
Credits lect+pc |
Assessment | Specialization | Course code lect/pc |
Semester | Status |
|---|---|---|---|---|---|---|
| 2 + 2 | 2 + 3 | exam + term grade |
pure math. | mm1c1vs3m mm1c2vs3m |
3 | compulsory |
Course coordinator
| Strong | Weak | Prerequisites | |
|---|---|---|---|
| Practice class | |||
|
Strong:
Discrete mathematics1L
(mm1c1vm1)
| |||
|
Strong:
| |||
| Lecture | |||
|
Weak:
a gyakorlat
| |||
Course objectives
Introduction of the basic notions and results of
discrete probability theory without measure theoretic
foundations.
Literature
Recommended:
- W. Feller: An Introduction to Probability Theory and Its Applications. Vol. 1, 3rd edition, Wiley, 1968.
Syllabus
- Probability space, conditional probablity, independence. Poincaré’s and Jordan’s formulae, the method of Rényi. Random variables, expectation, variance, covariance, correlation. Generating function.
- Simple symmetric random walk. Reflection principle and its applications. Distribution of random variables connected with the random walk: Position, times of recurrence, last visits and long leads, arcsine law, changes of sine, maxima and first passages, number of recurrences.
- Markov’s and Chebyshev’s inequalities, the weak law of large numbers. Convergence in probability. Moment generating function, rate function. Exponential inequalities (Chernoff-Hoeffding, Azuma).
- Asymptotic results for the binomial distribution. The DeMoivre-Laplace limit theorems, local and global versions. Limit distributions of random variables connected with the random walk. Poisson approximation via coupling, LeCam’s theorem.
- Almost sure convergence. The Borel-Cantelli lemma, the strong law of large numbers, Borel’s theorem. The law of iterated logarithm for symmetric Bernoulli variables. Lévy classes, EFKP theorem.
- Generating functions, connection with weak convergence. Random sums. Characterization of discrete infinitely divisible distributions. Point processes with independent and stationary increments. Branching processes, the probability of extinction.