Probability Problem 3.3 Critical Steps for solving it.

Prove that

$\mathbb{E}[X] = \int_{0}^{\infty}\mathbf{P}(X>x)dx - \int_{0}^{\infty}\mathbf{P}(X < -x) dx$.

Solution:

$\int_{0}^{\infty}P(X>x)dx

= \int_{0}^{\infty}\big(\int_{x}^{\infty}f_X(y)dy\big)dx

= \int_{0}^{\infty}\big(\int_{0}^{y}f_X(y)dx\big)dy

= \int_{0}^{\infty}f_X(y)\int_{0}^{y}dxdy

=\int_{0}^{\infty}yf_X{y}dy

$

The second step could be understood by drawing a graph on the x-y plane.