Mathematics BSc
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 +
pure math. mm1c1vs3m
mm1c2vs3m
3 compulsory
Strong Weak Prerequisites
Practice class
Strong:
Strong:
Analysis2L (mm1c1an2) or
Foundations of analysisL (mm1c1ap2)
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.