If f is a pdf, the integral of x*f(x) over the entire range where f(x) > 0 gives, of course, the expected value. Suppose that integrate the same function, x*f(x) from negative infinity up to t, giving a new function, G(t). So G(t) = E(x | x < t), and is measured in the same units as x, as the expected value is (while probability measures like the CDF are unitless).
I believe that G must be a well-known function with a name I should recognize and, on seeing it, clonk myself on the head and say “Of course!”. But I can not for the life of me think what it is. Am I right that this _is_a function with an established name? And if so, what is that name?
I have some problems to solve in this area and without a name it’s hard to search the literature. Help me out here, please.
Warmest regards, andrewH