Let $X \sim \operatorname{Gamma}(2,1)$, I would like to minimize with respect to $a$ $E|aX-1|=\int_0^{1/a}(1-ax)xe^{-x}dx+\int_{1/a}^\infty (ax-1)xe^{-x}dx$
Is there some neat way to do this? The only way I know is to use calculus on the RHS to find the minimum with respect to $a$. By neat, I mean a way that use facts from probability or gamma function? Thanks.