31 August 2015


$\E[X] = \int_{0}^\infty S(x) dx$

Proof:

$\int_{0}^\infty S(x) dx = \int_{x=0}^\infty \int_{t=x}^\infty f(t) dt dx$ $= \int_{x=0}^\infty \int_{t=0}^\infty f(t)I(t\geq x)dtdx$ $= \int_{t=0}^\infty f(t) \int_{x=0}^t dx dt = \int_0^\infty f(t) (t-0) dt = \int_0^\infty f(x)x dx = \E[X]$

(since f(t) is the PDF of the random variable $X$)

My study partners refer to this as the “Darth Vader Theorem” but I’m not sure where it comes from. Maybe because it is an evil trick that shows up in a lot of qualifying exam questions?