Let $p \in \left[ 1 , + \infty \right[$. Let $X$ be a positive and finite real random variable. Let $F _ { X }$ be the function defined for all $t \geq 0$ by
$$F _ { X } ( t ) = \mathbf { P } ( X > t ) .$$
Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { p - 1 } F _ { X } ( t ) \mathrm { d } t$ converges, then that
$$\mathbf { E } \left( X ^ { p } \right) = p \int _ { 0 } ^ { + \infty } t ^ { p - 1 } F _ { X } ( t ) \mathrm { d } t$$