1.8 If (a) = E|X a| < for some a, show that (a)...

1.8 If φ

(a) = E|X − a| < ∞ for some

a, show that φ

(a) is minimized by any median of X. [Hint: If m0 ≤ m ≤ m1 (in the notation of Problem 1.7) and m1 <

c, then E|X − c| − E|X − m| = (c − m)[P(X ≤ m) − P(X>m)] + 2 

m

(c − x)dP(x)].