The binomial distribution is one of the most important distributions in Probability and Statistics and serves as a model for several real life problems. Special cases of it were first derived by Pascal (1679) and Bernoulli (1713).
Definition and genesis. Denote by X the number of successes in a sequence of n ( ≥ 1) independent trials of an experiment, and assume that each trial results in a success (S) or a failure (F) with respective probabilities p (0 < p < 1) and q = 1 − p. The random variable (rv) X is said to have the binomial distribution with parametersn andp, and it is denoted by B(n, p). The probability mass function (pmf) f(x) of X is given by
where \(\left( {\begin{array}{*{20}c} n \\ x \\\end{array}} \right) = n!/x!(n - x)!\) for 0 ≤ x ≤ n and 0 otherwise.
In fact a typical element of the event {X = x} is a sequence SSFS…SF of xS’s and n − xF’s,...
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References and Further Reading
Bernoulli J (1713) Ars conjectandi. Thurnisius, Basilea
de Moivre A (1738) The doctrine of chances, 2nd edn. Woodfall, London
Laplace PS (1812) Théorie Analytique des Probabilités, 3rd edn. 1820, Courcier Imprimeur, Paris. Reprinted by EJ Gabay, 1992, Paris
Pascal B (1679) Varia opera Mathematica D. Petri de Fermat, Tolossae
Poisson SD (1837) Récherches sur la probabilité des jugements en matiere criminelle et en matiere civile, precedees des regles generales du calcul des probabilites, Paris: Bachelier, Imprimeur-Libraire pour les Mathematiques, la Physique, etc.
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Philippou, A.N., Antzoulakos, D.L. (2011). Binomial Distribution. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_146
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