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If $D$ and $D'$ are non-negative random variables, what are sufficient and/or necessary conditions for $E[D] \geq E[D']$ to imply $E[\min(c,D)]\geq E[\min(c,D')]$ for any $c\geq 0$.

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Although it is difficult to discern what is being asked here, one might be concerned with the identities $$ D=\int_0^{+\infty}[D\geqslant x]\cdot\mathrm dx \quad\text{and}\quad \min(c,D)=\int_0^{c}[D\geqslant x]\cdot\mathrm dx, $$ which yield $$ \mathrm E(D)=\int_0^{+\infty}\mathrm P(D\geqslant x)\cdot\mathrm dx \quad\text{and}\quad \mathrm E(\min(c,D))=\int_0^{c}\mathrm P(D\geqslant x)\cdot\mathrm dx. $$

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