Pascalovi procesi in njihova karakterizacija : delo diplomskega seminarja

Šenk, Benjamin (author), Bernik, Janez (mentor)

Diplomska naloga spada v področje teorije verjetnosti, ki se ukvarja s slučajnimi procesi. Na začetku si bomo pogledali osnove slučajnih procesov, homogeni in nehomogeni Poissonov proces ter markovske verige, nato pa bomo predstavili tri specifične procese: Yulov proces, Poissonov proces z V randomizirano intenziteto ter standardni Pascalov proces. Cilj diplomske naloge je dokazati enako porazdeljenost teh treh procesov. To poskušamo narediti kar po definiciji za enako porazdeljene procese ter z uporabo različnih matematičnih orodij. Na koncu predstavimo algoritme za izvajanje simulacij obravnavanih slučajnih procesov.

• Language: Slovenian
• Keywords: slučajni proces, Poissonov proces, markovske verige, porazdelitev slučajnega procesa, rojstni proces
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-161885
• UDC: 519.2
• COBISS.SI-ID: 208298499
• Publication date in RUL: 15.09.2024

Secondary language

• Language: English
• Title: Pascal processes and their characterization

Abstract

The thesis falls within the domain of probability theory, specifically focusing on stochastic processes. Initially, we will explore the fundamentals of stochastic processes, including homogeneous and non-homogeneous Poisson processes, as well as Markov chains. Subsequently, we will introduce three particular processes: the Yule process, the Poisson process with a random intensity V, and the standard Pascal process. The primary objective of this thesis is to demonstrate the equivalence in distribution of these three processes. We aim to achieve this directly from the definition for equally distributed processes and by employing a variety of mathematical tools. Finally, we present algorithms for performing simulations of the discussed stohastic processes.

• Keywords: stohastic process, Poisson process, Markov chains, distribution of a stohastic process, birth process