Talk:Probability box

Three formulations of the constraints expressed by a p-box

There are different ways (shown below) to formulate the displayed constraints in the section 'Mathematical definition'. Each seems to have disadvantages. A formulation based on Riemann-Stieltjes integrals introduces the notation dF(x), which will likely be foreign to a lot of readers. We could formulate the integrals in terms of the quasi-inverse of the distribution function, but then we'd need to explain quasi-inverses. Or, we could use a formulation suggested by the entry Variance#Calculation from the CDF, but that requires identifying the minimum possible value of the random x-value. Currently, the entry uses the first of these options. Does anyone have suggestions about how to express the constraints?

Scwarebang (talk) 05:00, 31 March 2011 (UTC)

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F (x) ≤ F(x) ≤ F(x),

∫ ^∞
_-∞ x dF(x) ∈ m,

∫^∞
_-∞ x²dF(x)) – (∫ ^∞
_-∞ x dF(x))² ∈ v, and

F ∈ F.

These Riemann-Stieltjes integrals do not depend on the differentiability of F.

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F (x) ≤ F(x) ≤ F(x),

∫ ¹
₀ F ^-1(u)du ∈ m,

∫ ¹
₀ F ^-1(u)²du – (∫ ¹
₀ F ^-1(u)du)² ∈ v, and

F ∈ F

where F ^-1 is the quasi-inverse of F. These formulations do not depend on the invertibility of F but only on its monotonicity.

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F (x) ≤ F(x) ≤ F(x),

∫ ^∞
_a (1 – F(x – a))dx ∈ m,

2∫^∞
_a (x – a)(1 – F(x – a))dx – (∫ ^∞
_a (1 –F(x – a))dx)² ∈ v, and

F ∈ F

where a = sup_{F(x)=0 x is the smallest possible value of the random variable.}

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