Teoria Statistica Delle Classi e Calcolo Delle Probabilit&amp;#224;

doi:10.4135/9781412961288

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Teoria Statistica Delle Classi e Calcolo Delle Probabilità

Edited by:
Neil J. Salkind
In:Encyclopedia of Research Design
Chapter DOI:https://doi.org/10.4135/9781412961288.n455
Subject:Research Design
Keywords:probability; probability theory

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Carlo Emilio Bonferroni was an Italian mathematician, who is known in the field of statistical simultaneous inference for the so-called Bonferroni inequalities that Bonferroni described in his book Teoria Statistica Delle Classi e Calcolo Delle Prob-abilità [Statistical Class Theory and Calculation of Probability]. This work has made significant contributions to probability theory and statistical inference.

The book, divided into three sections, is a systematic study of probability theory and its applications on set (class) theory. In the introduction, Bonferroni states that the probability of an event is a primitive physic magnitude that does not have a definition; however, it is possible to explain it in some postulate or axiom. The classical postulate assumes that if an event is separated by other incompatible, complementary, and equally possible events, then the probability of that event is the ratio between the number of cases favorable to it and the number of all events possible.

Relationship among Probabilities

In the first section, Bonferroni, before enouncing his inequalities, describes the following probabilities and their relationship:

Simultaneous probabilities: In a set composed of m objects, if n characteristics C1, C2… … Cn are considered and indicated by mi, the number of objects having the characteristic Ci, by mij the number of objects having simultaneously CiCj and by mijh those having CiCjCh etc… then the probabilities are

Contrary probabilities:

is the contrary probability to pij …k, that is the probability of an object without having the characteristics Ci, Cj,…, Ck. The probability is

and in more general

then it is possible to write

Exclusive probability: A probability that an object has some characteristics excluding the others. It is defined by the product between the probabilities of those characteristics taken into consideration and the contrary probabilities.

r-exact multiplicity: The multiplicity of an object is defined as the number of characteristics that it possesses, from 0 to n.

r-minimum multiplicity: An object that possesses at least r characteristics.

r-maximum multiplicity: An object that possesses at most r characteristics, that is 0, 1,2,… or r.

Incompatible characteristics, complementary characteristics: The characteristics Ci are incompatible when an object has at most one; for CiCj, pij = 0. They are complementary if an object possesses at least one. The formula is

The characteristics are incompatible and complementary when 1 − S1 = 0; that is, ∑p = 1.

Contrary characteristics and duality law: If m objects are described by their contrary “not C” characteristics rather than by the characteristics Ci, the simultaneous probabilities are qi, qij, qijh, … and are opposites of the affirmative probabilities pi, pij, pijh, … For the duality law, each calculation developed on the affirmative probabilities is reproducible on the contrary probabilities.