Find an expression for $\int_{-\infty}^{0}xf_X(x) dx$ in terms of CDF

Question

I know $E(X) = \int^{\infty}_{0}xf_X(x) dx$ when the range is $\mathbb{R}^+$, but how would I do it for the opposite range - I tried a change of variables $u = -x$ but when I swap my order of integration it all goes wrong.

Answer 1

Just integrate by parts obtaining

$$\mathbb{E}[X]=\int_{-\infty}^0xf_X(x)dx=\left. x F_X(x)\right]_{-\infty}^0-\int_{-\infty}^0F_X(x)dx=-\int_{-\infty}^0F_X(x)dx$$

the first addend is zero. You can verify it simply using de l'Hôpital

In any good Statistics texbook you will find a definition of mean that is

$$\mathbb{E}[X]=\int_0^{\infty}[1-F_X(x)]dx-\int_{-\infty}^0F_X(x)dx$$