Ujemanje vsote pik kock z enakomerno porazdelitvijo

delo diplomskega seminarja

Anamarija Potokar (Author), Roman Drnovšek (Mentor)

Abstract

V diplomski nalogi obravnavamo problematiko porazdelitve vsote pik $n$ kock z $m$ stranicami in njeno približevanje enakomerni porazdelitvi. Najprej za primer dveh kock s šestimi stranicami s pomočjo rodovnih funkcij pokažemo, da enakomerna porazdelitev slučajne spremenljivke, ki predstavlja vsoto padlih pik, ni mogoča. Nato rezultat posplošimo na primer $n$ kock z $m$ stranicami, kjer dokaz z rodovnimi funkcijami odpove, zato do odgovora pridemo po drugi poti, preko izreka. V drugem delu naloge se ukvarjamo s tem, kako se enakomerni porazdelitvi vsote pik dveh kock z m stranicami karseda približamo z uporabo optimalnih kock, kjer ugotovimo, da sta optimalni kocki simetrični, a za $m > 2$ nista identični. Potem dokažemo, da je v teoretični situaciji, ko dopuščamo negativne verjetnosti, enakomerna porazdelitev vsote mogoča. Na koncu s pomočjo kode, implementirane v programskem jeziku Python, preverimo veljavnost izreka o optimalnih kockah, nato pa kodo uporabimo še za preizkušanje splošnih primerov za poljubna $n$ in $m$.

Keywords

vsota pik kock;enakomerna porazdelitev;rodovna funkcija;optimalni kocki;simetrični kocki;

Data

Language:	Slovenian
Year of publishing:	2025
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[A. Potokar]
UDC:	519.2
COBISS:	246906883
Views:	104
Downloads:	27
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Matching dice sum to a uniform distribution
Secondary abstract:	This thesis explores the problem of the distribution of the sum of $m$-sided dice, and how close this distribution can be to a uniform distribution. First, we show using generating functions that a uniform distribution of the random variable representing the sum of two six-sided dice is not possible. We then generalize the result to the case of $n$ dice with $m$ sides. Since the proof using generating functions fails in this case, we take a different approach via a theorem. In the second part of the thesis, we explore how to approximate the uniform distribution of the sum of two $m$-sided dice as closely as possible using optimal dice. We find that the optimal dice are symmetric, but for $m > 2$ they are not identical. We then prove that in a theoretical scenario where negative probabilities are allowed, a uniform distribution of the sum is possible. Finally, using Python code, we verify the validity of the theorem about two optimal dice, and then use the code to test general cases for arbitrary $n$ and $m$.
Secondary keywords:	sum of dice;uniform distribution;generating function;optimal dice;symmetric dice;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages:	27 str.
ID:	27245471