Expectation through survival function

Question

I understand how we get $$E[x] = \int_0^\infty (1-F(x)) dx$$ but why does this apply only to non-negative random variables

Answer 1

Consider the integral, and integrate partially: $$\int\limits_{-\infty}^0 1-F(x)\,dx=\left[ x(1-F(x)) \right]_{-\infty}^0+\int\limits_{-\infty}^0 xf(x)\,dx$$ In this formula on the right side, if $X\in L^1$ the integral is finite, but the change of the product is infinite. Informally $$\int\limits_{-\infty}^\infty 1-F(x)\,dx=\infty + EX $$