Compute the conditional expectation of a standard normal random variable X given X>a

Question

$X$ is a standard normal random variable with density $\phi(x)=\frac{1}{\sqrt{2\pi}}exp(-\frac{x^2}{2})$ and $a$ is a given constant. Denote $p=P[X>a]$.

My question is how to derive $E[X|X>a]=\frac{\phi(a)}{p}$.

Answer 1

hint

This is equivalent to showing $$ \int_a^\infty x\phi(x)dx = \phi(a).$$ From here, use the fact that $\phi'(x) = -x\phi(x).$