Asymptotic expansions of n!: from Stirling to Gosper and beyond
Combinatorics: number of onto mappings, Stirling numbers and
Bell numbers
Combinatorics: classical questions around the cardinal of random
sets
Probability and arithmetics: the probability to draw coprime
numbers
The geometric distribution and how it can help to park your car
The problem of points: Pascal, Fermat and the Chevalier de Méré
A short promenade into the world of random walks: around
Polya's theorem
Borel-Cantelli's lemmas: theoretical and practical applications
Almost sure convergence vs. convergence in probability: some
niceties
Uniform integrability: main theorems and a result by de la
Vallée Poussin
Convergence in distribution: from portmanteau to Slutsky and
beyond
Dynkin's pi-lambda theorem and Kolmogorov's zero-one law
From weak to strong: around the law of large numbers
Series of random variables: Kolmogorov's three-series theorem
A simple CLT: de Moivre-Laplace theorem. Application to
statistics
The characteristic function: injectivity, Lévy's continuity
theorem, and the CLT
Gaussian vectors, Cochran's theorem, and applications to
statistics
The coupon collector's problem: from moments to asymptotic analysis
The characteristic function: regularity and moments
A short introduction to the moment problem
Kalman filtering: Bayesian approach and optimization approach