Probability bounds analysis: Difference between revisions

Revision as of 06:07, 5 July 2015

Revision as of 09:29, 20 April 2016

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===Mathematical details===

Let {{Unicode|&#x1D53B;}} denote the space of distribution functions on the [[real number]]s {{Unicode|ℝ}}, i.e., {{Unicode|&#x1D53B;}} = {''D'' | ''D'' : {{Unicode|ℝ}} → [0,1], ''D''(''x'') ≤ ''D''(''y'') whenever ''x'' < ''y'', for all ''x'', ''y'' [[Naive set theory#Sets.2C membership and equality|∈]] {{Unicode|ℝ}}}, and let {{Unicode|&#x1D540;}} denote the set of real [[Interval (mathematics)|intervals]], i.e., {{Unicode|&#x1D540;}} = {''i'' | ''i'' = [''i''₁, ''i''₂], ''i''₁ ≤ ''i''₂, ''i''₁, ''i''₂ ∈ {{Unicode|ℝ}}}. Then a p-box is a quintuple {''{{overbar|F}}'', <u>''F''</u>, ''m'', ''v'', '''F'''}, where ''{{overbar|F}}'', <u>''F''</u> ∈ {{Unicode|&#x1D53B;}}, while ''m'', ''v'' ∈ {{Unicode|&#x1D540;}}, and '''F''' ⊆ {{Unicode|&#x1D53B;}}. This quintuple denotes the set of distribution functions ''F'' ∈ '''F''' ⊆ {{Unicode|&#x1D53B;}} such that ''{{overbar|F}}''(''x'') ≤ ''F''(''x'') ≤ <u>''F''</u>(''x'') for all ''x'' ∈ {{Unicode|ℝ}}}, and the mean and variance of ''F'' are in the intervals ''m'' and ''v'' respectively.

If ''F'' is a [[distribution function]] and ''B'' is a [[p-box]], the notation ''F'' ∈ ''B'' means that ''F'' is an

element of ''B'' = {''B''₁, ''B''₂, [''m''₁,''m''₂],

[''v''₁,''v''₂], '''B'''}, that is,

''B''₂(''x'') ≤ ''F''(''x'') ≤ ''B''₁(''x''), for all ''x'' ∈ {{Unicode|ℝ}},

[[Expected value|E]](''F'') ∈ [''m''₁,''m''₂],

[[Variance|V]](''F'') ∈ [''v''₁,''v''₂], and

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that of intervals on the real line, except that the endpoints are distributions rather than points.

The notation ''X'' ~ ''F'' denotes the fact that ''X''∈{{Unicode|ℝ}}∈ℝ is a random variable governed by the

distribution function ''F'', that is, ''F'' = ''F''(''x''):{{Unicode|ℝ}}→ℝ→[0,1]:x→Pr(''X''≤''x'').

<!-- can I get the "mapsto" character without resorting to ugly <math> ? -->

Let us generalize the tilde notation for use with p-boxes. We will write