Derivation of a well-known property of the standard normal distribution I have found the following property of the standard normal distribution:
$ \int_r^\infty xf(x) dx  = f(r) $
where $f(.)$ is the pmf of the standard normal distribution. 
$f(r)$ and $F(r)$ are defined as follows:
$f(r) = \cfrac{1}{\sqrt {2π}} e^{-r^2/2} $ ; $ \int_r^\infty f(x) dx $
$ F(.) $  is the cdf of the standard normal distribution
Does anybody knows how this property is derived or where I can find this derivation?
Steven

It is obtained by just noting that $-xe^{-x^{2}/2}$ is the derivative of $e^{-x^{2}/2}$, so $\int_r^{\infty} xf(x)dx =f(r)$.
