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, 3^{rd} 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 (ChernoffHoeffding, Azuma).
 Asymptotic results for the binomial distribution. The DeMoivreLaplace 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 BorelCantelli 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.