Discrete Case
Discrete probability theory needs only at most countable sample spaces Ω. Probabilities can be ascribed to points of Ω by the probability mass function p: Ω→ such that ∑ω∈Ω p(ω) = 1. All subsets of Ω can be treated as events (thus, = 2Ω is the power set). The probability measure takes the simple form
The greatest σ-algebra = 2Ω describes the complete information. In general, a σ-algebra ⊆ 2Ω corresponds to a finite or countable partition Ω = B1 ⊔ B2 ⊔ ..., the general form of an event A ∈ being A = Bk1 ⊔ Bk2 ⊔ ... (here ⊔ means the disjoint union.) See also the examples.
The case p(ω) = 0 is permitted by the definition, but rarely used, since such ω can safely be excluded from the sample space.
Read more about this topic: Probability Space
Famous quotes containing the words discrete and/or case:
“We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the child’s life, these words are discrete and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”
—Selma H. Fraiberg (20th century)
“The bond between a man and his profession is similar to that which ties him to his country; it is just as complex, often ambivalent, and in general it is understood completely only when it is broken: by exile or emigration in the case of one’s country, by retirement in the case of a trade or profession.”
—Primo Levi (1919–1987)