Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution.


I would like to ask about the following specific way to describe the distribution in terms of the variance of tails.


Given a number t between 0 and 1 let s be such that the probability that $X\ge s$ equals t. (In other words, $\int\_s^{\infty}f(x)=t$.) Let $W\_X(t)$ be the variance of $X$ conditioned on $X\ge s$.


The function $W\_X(t)$ seems an interesting way to describe the original probability distribution, and its tail behavior. I wonder if there is an explicit useful way to move from $W$ back to $f$; Can you give a dictionary between the tail behavior of $f$ and of that of $W$; And an explicit description of $W$ for some "famous" probability distributions. (E.g. the normal distribution, and the Tracy Widom distribution.) And of course if this kind of "transform" is considered in the literature. 


**UPDATE:** Many thanks to Robert and Brendan to their answers. Robert's asymptotic expansion seem amazing. I will be happy to understand the asymptotics as t goes to 0.


Looking at various distribution one sees that when the distribution is exponential, W is constant; when the distribution is normal and thus decays doubly exponentially, W(t) behaves like $\log^{-1}(1/t)$. I am curious how the asymptotic decay of $W(t)$ translates to the tail behavior of the probability distribution. For the tracy Widom distribution when you ask what is the probability that $X \ge M+t \sigma$ you get an expression like $e^{-t^{3/2}}$ and for the probability that $X \le M-t \sigma$ you get something like $e^{-t^3}$. I wonder how these tails behavior can be described in terms of $W\_X(t)$ and $W'\_X(t)$ respectively, where $W'\_X(t)$ deals with conditioning on $X \le s$ rather than on $X \ge s$. So let me form a simple question:


**Question:** If $W\_X(t)$ behaves like $\log^a (1/t)$, for some real $a>0$, what does it say about the tail behavior of $X$. (More generally, how the decay of $W$ translates to the decay of the distribution.) 



A Variance-Tail Description for Continuous Probability Distributions
For a normal distribution with mean $0$ and standard deviation $\sigma$, I get (with help from Maple)
$$W\_X(t) = \sigma^2 + \sqrt{\frac{2}{\pi}} \frac{\sigma t}{1 - \text{erf}(t/(\sqrt{2}\sigma))}
e^{-t^2/(2 \sigma^2)} + \frac{2 \sigma^2}{\pi (1 - \text{erf}(t/(\sqrt{2}\sigma))^2}
e^{-t^2/\sigma^2}$$


As $t \to \infty$, this rather fearsome expression has the asymptotic series


$$W\_X(t) \sim {\frac {{{\sigma}}^{4}}{{t}^{2}}}-6\ {\frac {{{\sigma}}^{6}}{{t}
^{4}}}+50\ {\frac {{{\sigma}}^{8}}{{t}^{6}}}-518\ {\frac {{{\sigma}}^{10}}{{t}^{8}}}+6354\ {\frac {{{\sigma}}^{12}}{{t}^{10}}}-
89782\ {\frac {{{\sigma}}^{14}}{{t}^{12}}}+1435330\ {\frac {{{\sigma}}^{16}}{{t}^{14}}}-25625910\ {\frac {{{\sigma}}^{18}}{{t}^{16
}}}+ \ldots$$


where the coefficients appear to be the entries of OEIS sequence A005416 <https://oeis.org/A005416> which is described as "Vertex diagrams of order 2n".


