Is there a name for $\operatorname{median}(xf(x))$ where $f(x)$ is a probability density function on real numbers (or positive real numbers)? If yes, what are the properties of this number?

This question arose from a real world problem, where I have a sample of some value $\{x_n\}$, and I'm interested in finding $a$ that divides $\sum_{n} x_n$ in half, i.e. $a$ such that:

$$\sum_{x_n<a}x_n = \sum_{x_n\geq a}x_n = \frac{1}{2}\sum_{n}x_n.$$



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