Boole's inequality

Boole's inequality

In probability theory, Boole's inequality, named after George Boole, (also known as the union bound) says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events.

Formally, for a countable set of events "A"1, "A"2, "A"3, ..., we have

:Prleft [igcup_{i} A_i ight] leq sum_i Prleft [A_i ight] .

In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is "σ"-sub-additive.

Bonferroni inequalities

Boole's inequality may be generalised to find upper and lower bounds, known as Bonferroni inequalities, on the probability of finite unions of events.

Define::S_1 := sum_{i=1}^n Pr(A_i),::S_2 := sum_{iand for 2 < "k" ≤ "n",::S_k := sum Pr(A_{i_1}cap cdots cap A_{i_k} ),where the summation is taken over all "k"-tuples of distinct integers.

Then, for odd "k" ≥ 1,::Prleft( igcup_{i=1}^n A_i ight) leq sum_{j=1}^k (-1)^{j+1} S_j,and for even "k" ≥ 2,::Prleft( igcup_{i=1}^n A_i ight) geq sum_{j=1}^k (-1)^{j+1} S_j.

Boole's inequality is recovered by setting "k" = 1.

When "k" = "n", then equality holds and the resulting identity is the inclusion-exclusion principle.

References

ee also

* Inclusion-exclusion principle
* Carlo Emilio Bonferroni


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Boole's inequality — Math. the theorem that the probability of several events occurring is less than or equal to the sum of the probabilities of each event occurring. [named after G. BOOLE] * * * …   Universalium

  • Boole's inequality — Math. the theorem that the probability of several events occurring is less than or equal to the sum of the probabilities of each event occurring. [named after G. BOOLE] …   Useful english dictionary

  • Inequality — In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not (See also: equality) *The notation a < b means that a is less than b . *The notation a > b means that a is… …   Wikipedia

  • Inequality (mathematics) — Not to be confused with Inequation. Less than and Greater than redirect here. For the use of the < and > signs as punctuation, see Bracket. More than redirects here. For the UK insurance brand, see RSA Insurance Group. The feasible regions… …   Wikipedia

  • Inegalite de Boole — Inégalité de Boole En théorie des probabilités, l inégalité de Boole stipule que pour toute famille finie ou dénombrable d événements, la probabilité que l un au moins des événements se réalise est inférieure ou égale à la somme des probabilités… …   Wikipédia en Français

  • Inégalité de Boole — En théorie des probabilités, l inégalité de Boole stipule que pour toute famille finie ou dénombrable d événements, la probabilité que l un au moins des événements se réalise est inférieure ou égale à la somme des probabilités des événements pris …   Wikipédia en Français

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • List of inequalities — This page lists Wikipedia articles about named mathematical inequalities. Inequalities in pure mathematics =Analysis= * Askey–Gasper inequality * Bernoulli s inequality * Bernstein s inequality (mathematical analysis) * Bessel s inequality *… …   Wikipedia

  • Multiple comparisons — In statistics, the multiple comparisons or multiple testing problem occurs when one considers a set of statistical inferences simultaneously.[1] Errors in inference, including confidence intervals that fail to include their corresponding… …   Wikipedia

  • List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”