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pr.likelihood – A practical equation moving the inverse duty Answer

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pr.likelihood – A practical equation moving the inverse duty

$newcommandepepsilonnewcommandR{mathbb R}$Let $P$ denote the clique of all steady likelihood density capabilities (pdf’s) $p$ on $R$ vanishing at $pminfty$. Let us say {that a} pdf $pin P$ is good if for every diminutive sufficient $ep>0$ the practical equation
$$g(x)-g^{-1}(x)=ep, p(x)quadforall xinRtag1$$
has an answer $gcolonRtoR$, which is an growing steady duty such that $g(x)>x$ for all actual $x$.

It is limpid that, if a pdf $pin P$ is sweet, then for any actual $a$ and any actual $b>0$ the pdf $p_{a,b}$ given by the components $p_{a,b}(x):=b,p(a+bx)$ for actual $x$ is sweet as effectively.

The drawback right here is to picture the clique of all good pdf’s $pin P$.

Of passage, there may be all the time a tautological characterization: a pdf $pin P$ is sweet if and solely whether it is good. Any non-tautological characterization would breathe of curiosity, together with incomplete ones, comparable to circumstances which can be solely adequate or solely needful for the goodness. In specific, it might breathe of curiosity to know if the “triangular” pdf $p_triangle$ given by the components $p_triangle(x):=max(0,1-|x|)$ for actual $x$ is sweet.

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