In the diploma thesis, we learn about the Bell polynomials, the role they play in combinatorics, and the formulas for their computation, explicit and recursive ones alike. We also develop a special tool that sheds new light onto the Bell polynomials: the umbral calculus and the closely related umbral algebra. We then use the Bell polynomials to derive the identities that connect two important concepts from the study of probability: the moments and the cumulants. Finally, we derive Faà di Bruno's formula for higher derivatives of a composite function and characterize the polynomials of binomial type.
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