Gaussovské normální rozdělení

Julian Russell WIRAHADIKUSUMA

Bakalářská práce

Gaussian normal distributions

Abstract:

This thesis rigorously investigates the Gaussian normal distribution, denoted as X~N(?,?^2), where ? represents the mean and ?^2 denotes the variance. The Gaussian distribution, a fundamental construct in probability theory, is distinguished by its properties of symmetry, central tendency, variance, skewness, kurtosis, and higher-order moments. The inherent characteristics of the Gaussian distribution facilitate its application in a plethora of theoretical and practical contexts, underpinning the analysis of various stochastic processes. Through an exploration of the Central Limit Theorem, this work elucidates the convergence behaviour of the sum of independent and identically distributed random variables towards the standard normal distribution, N(0,1), upon standardization. This pivotal theorem underscores the Gaussian distribution's role in approximating other significant distributions such as the Binomial under appropriate conditions. Employing characteristic functions and moment-generating functions, we provide rigorous proofs and derivations that solidify the Gaussian distribution's foundational status in statistical theory.

Abstract:

This thesis rigorously investigates the Gaussian normal distribution, denoted as X~N(?,?^2), where ? represents the mean and ?^2 denotes the variance. The Gaussian distribution, a fundamental construct in probability theory, is distinguished by its properties of symmetry, central tendency, variance, skewness, kurtosis, and higher-order moments. The inherent characteristics of the Gaussian distribution facilitate its application in a plethora of theoretical and practical contexts, underpinning the analysis of various stochastic processes. Through an exploration of the Central Limit Theorem, this work elucidates the convergence behaviour of the sum of independent and identically distributed random variables towards the standard normal distribution, N(0,1), upon standardization. This pivotal theorem underscores the Gaussian distribution's role in approximating other significant distributions such as the Binomial under appropriate conditions. Employing characteristic functions and moment-generating functions, we provide rigorous proofs and derivations that solidify the Gaussian distribution's foundational status in statistical theory.

Keywords

Gaussian distribution, moment-generating functions, characteristic function, skewness, kurtosis, Central Limit Theorem

Jazyk práce: angličtina

Datum vytvoření / odevzdání či podání práce: 30. 7. 2024

Zveřejnit od: 31. 12. 2999

Obhajoba závěrečné práce

• Obhajoba proběhla 21. 8. 2024
• Vedoucí: Zhivorad Tomovski, PhD.

OSTRAVSKÁ UNIVERZITA

Přírodovědecká fakulta

Bakalářský studijní program / obor:
Mathematics / Mathematics