Name of the functional $p \mapsto P_X[p] = X\int_X^\infty p(x)\,dx$ Let $p(x): \mathbb{R} \rightarrow \mathbb{R}^+_0$ be a probability distribution with $\int_{-\infty}^\infty p(x)\,dx = 1$. Is there a special name for the parametrized functional
$$p \mapsto P_X[p] = X\int_X^\infty p(x)\,dx\ ?$$
The name of $p \mapsto I_X[p] = \int_X^\infty p(x)\,dx$ is just the definite integral of $p$ from $X$ to infinity and $p \mapsto Id_X[p] = X$ possibly doesn't have a name due to its triviality. But what about the product $X\int_X^\infty p(x)\,dx$?
Whether there is a special name or not: Which use cases are there for this functional, e.g. in physics, economics, or statistics? I came across it when I tried to calculate the expected revenue when selling a whole stock of products for a given price $X$ per item with $p(x)$ the distribution of willingness to pay.
Note the difference with the expected value
$$p \mapsto \mathbb{E}[p] = \int_{-\infty}^\infty x\ p(x)\,dx,$$
which is just another functional.

To show what the functional may be good for, I plotted it (blue) for a number of distributions $p(x)$ of willingness to pay (green). The maximum is at the price $X$ for which a maximal revenue can be expected. Depicted in red is the reservation price of the supplier (willingness to accept, set to $1$).

 A: Not a complete answer, just too big for a comment.
Let me put this functional in a simpler form: $P_x(\mu) = x\cdot(1 - F_\mu(x))$ where $F_\mu$ is the CDF for a probability distribution $\mu$. I have not seen this functional being used with a particular name anywhere, but I definitely expect this to be met in economics-like disciplines due to a hard cut and saturation of value at $x$. That looks extremely artificial to me, though by no means I'm an expert say in physics. Regarding economics, I would expect this to be used in at least two cases

*

*Options pricing. In digital-like options $P_x(\mu)$ would be a price of an option with a strike $x$ if the underlying distribution at expiry is $\mu$.


*Mechanism design/auction theory. There it is important to consider both the probability that one gets the lot, and the value of its lot to the winner.


*(added with an edit) Ok, after you've edited I've immediately recalled that it is also used in the proof of Markov's inequality, hence likely be used in probability bounds estimations. For example, in the ruin theory - another area combining economics and probability - once often aims at deriving upper bounds on the ruin probability for a high value of the initial capital. Some of those relate to the aforementioned estimation, hence likely using your operator. Perhaps, my answer just describes at least three uses of the H-PS operator.
