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I'm trying to use Mathematica to solve a problem involving any continuous probability distribution $F(t)$ such that:

$$\begin{align*} F(t)&=0, \; t\leq0\\ F(t)&=1, \; t\geq1 \end{align*}$$

For example, I'd like to evaluate the following expression:

$$\mathbb E[(A - F(t))\mathbf I_{F(t)<A}]$$

where $\mathbb E$ is the expectation operator, $A$ is a constant, and $\mathbf I$ is the indicator function.

My question is: how do I evaluate the expression above without assuming a particular probability distribution? I've searched here and elsewhere but wasn't able to find anything that I could use. (Although Expectation of a family of random variables seems related.)

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  • $\begingroup$ Thank you, @rhermans for the comment, and J. M. for markdown help. I'll come back soon to update my question. $\endgroup$ – Houston Mooncat Aug 24 '17 at 18:49
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This might be considered an answer in that there doesn't appear to be a general solution that results in a function that just depends on $A$ (assuming that's what you mean by not having to specify any particular distribution other than meeting the given constraints).

For the two specific distributions I've tried (beta and logit-normal), the answer for $A>1$ is $A-\frac{1}{2}$. So that might possibly be a general solution. (I don't know how to prove that one way or the other.)

But for $0<A<1$, the results don't immediately look generalizable:

Beta distribution

a =.
α =.;
β =.;
Integrate[(a - CDF[BetaDistribution[α, β], t]) PDF[BetaDistribution[α, β], t], {t, 0, a},
  Assumptions -> {α > 0, β > 0, 0 < a < 1}]

$\frac{2 a \Gamma (\alpha )^2 \Gamma (\beta )^2 B_a(\alpha ,\beta )-\Gamma (\alpha +\beta )^2 B(\alpha ,\beta ) B_a(\alpha ,\beta ){}^2}{2 \Gamma (\alpha )^2 \Gamma (\beta )^2 B(\alpha ,\beta )}$

Logit-normal distribution

a =.
μ =.;
σ =.;
f[t_, μ_, σ_] := Exp[-(Log[t/(1 - t)] - μ)^2/(2 σ^2)]/(σ t (1 - t) (2 π)^(1/2))
F[t_, μ_, σ_] := (1 + Erf[(Log[t/(1 - t)] - μ)/(σ 2^(1/2))])/2
Integrate[(a - F[t, μ, σ]) f[t, μ, σ], {t, 0, a}, 
  Assumptions -> {0 < a < 1, σ > 0, μ ∈ Reals}]

$\frac{1}{8} \left(\text{erf}\left(\frac{\log \left(\frac{1}{a}-1\right)+\mu }{\sqrt{2} \sigma }\right)+4 a-1\right) \text{erfc}\left(\frac{\log \left(\frac{1}{a}-1\right)+\mu }{\sqrt{2} \sigma }\right)$

If those results are evaluated at the same value of $A$ but different values of the parameters, one does not (always) obtain the same result:

Beta distribution

g[a_, α_, β_] := (2 a Beta[a, α, β] Gamma[α]^2 Gamma[β]^2 - Beta[α, β] Beta[
    a, α, β]^2 Gamma[α + β]^2)/
(2 Beta[α, β] Gamma[α]^2 Gamma[β]^2)
g[1/2, 1, 1]
(* 1/8 *)
g[1/2, 3, 4]
(* 231/2048 *)

Logit-normal distribution

g[a_, μ_, σ_] := 1/8 (-1 + 4 a + Erf[(μ + Log[-1 + 1/a])/(Sqrt[2] σ)])
  Erfc[(μ + Log[-1 + 1/a])/(Sqrt[2] σ)]
g[1/2, 0, 1]
(* 1/8 *)
g[1/2, 1, 2]
(* 1/8 (1+Erf[1/(2 Sqrt[2])]) Erfc[1/(2 Sqrt[2])] *)

In short, I think you'll need to specify the distribution (but potentially get a symbolic result in terms of the distributions parameters).

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