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