Expectation and Distribution Function? Consider X as a random variable with distribution function $F(x)$. Also assume that $|E(x)| < \infty$. the goal is to show that for any constant $c$, we have:
$$\int_{-\infty}^{\infty} x (F(x + c) - F(x)) dx = cE(X) - c^2/2$$
Does anyone have any hint on how to approach this?
Thanks

Here is an approach valid for every distribution: integrate the pointwise identity
$$
\int_{-\infty}^{+\infty}x\,\mathbf 1_{x\lt X\leqslant x+c}\,\mathrm dx=\int_{X-c}^{X}x\,\mathrm dx=cX-c^2/2.
$$
