20.500.12556/DKUM-12614 VEJITVENI PROCESI BRANCHING PROCESSES Vejitveni proces je matematični model razvoja in rasti neke populacije. Opazujemo lahko razvoj populacije po generacijah ali pa po časovnih intervalih. Takim vejitvenim procesom pravimo časovno odvisni. Obravnavali smo jih s pomočjo rodovne funkcije in s pomočjo markovskih verig. Najprej so razloženi osnovni pojmi, kot so neodvisne naključne spremenljivke, rodovna funkcija in markovske verige, ki so potrebni za razumevanje teorije o vejitvenih procesih. Pri tem gre predvsem za izračun verjetnosti izumrtja populacije. Poleg teoretične obravnave izumrtja, je na konkretnih primerih predstavljen vejitveni proces z začetno geometrijsko porazdelitvijo, za katerega je narejena tudi računalniška simulacija. Program je narejen v programskem jeziku C# in vejitveni proces simulira z gradnjo drevesa in izračunom verjetnosti izumrtja. Simulacija da rezultate, ki so primerljivi s teoretično izračunanimi. The branching process is a mathematical model of development and growth of a population. We can suppose that a population evolves in generations or we attach another random variable, called ’age’. We name them age-dependent branching processes. For describing such processes we use probability generating functions and markov chains. At …rst are explained basic concepts, such as dicrete random variables, proba- bility generating functions and markov chains, which are necessary to understand the theory of branching processes. Especially we are interested in expressing proba- bility of ultimate extinction. In addition to the theoretical treatment of extinction, are also exemples which presents the geometrical distributed processes. Those are included in computer simulation. The program, designed in C #, simulate growth of a population with family tree of a branching process and calculate the probability of extinction. Results of the simulation are comparable to the theoretical results. vejitveni proces µcasovno odvisen vejitveni proces rodovna fun- kcija markovska veriga branching process age-dependent branching process probability gen- erating function markov chain true true false [M. Kramer] Slovenski jezik Angleški jezik Diplomsko delo 2009-12-02 17:55:44 2009-12-23 12:35:11 2022-04-12 07:38:53 2022-04-20 03:06:44 2009 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 51(043.2) 17314568 URN:SI:UM:DK:SXJJPW5W UNI_Kramer_Mojca_2009.pdf UNI_Kramer_Mojca_2009.pdf 1 9F26C7127766D91B9C75DF242F6E9A72 0df674af7439fcda2fccb638d04cc3df15ed010d4072f63a2d9997472950759b 7693e914-7c03-11eb-bb7a-00155d0001ca 20.500.12556/dkum/44670d17-19c9-4872-b465-dbb0c3f3fc07 https://dk.um.si/Dokument.php?lang=slv&id=12188 Fakulteta za naravoslovje in matematiko 0 0 0