Possible Duplicate:
Why does the median minimize $E(|X-c|)$?
Can someone tell how to calculate the $y$ of $\min E[|X-y|]$,where $X$ has a continuous density function $f(x)$?
Possible Duplicate:
Why does the median minimize $E(|X-c|)$?
Can someone tell how to calculate the $y$ of $\min E[|X-y|]$,where $X$ has a continuous density function $f(x)$?
If the expectation exists, $y$ minimizes your expression if and only if $y$ is a median of $X$.
From a calculation point of view, you are then solving $\int_{-\infty}^y f_X(t)\,dt=\frac{1}{2},$ where $f_X(t)$ is the density function of $X$.
Proofs can be found in many places. For example, you can find the proof of a more general result on Math Stack Exchange.