See Barron (1991). (a) Show that for any random variable V with values in some interval of length B one has                                           Var{V } ≤ B2 4 . (b) Show that the inequality...

See Barron (1991).

(a) Show that for any random variable V with values in some interval of length B one has

                                          Var{V } ≤ B2 4 .

(b) Show that the inequality (7.7) can be improved as follows: Assume |Y | ≤ L a.s. and let f be a function f : Rd → [−L, L]. Set

                                           Z = |f(X) − Y | 2 − |m(X) − Y | 2 .

Then

                                          σ2 = Var{Z} ≤ 8L2 E{Z}.

Hint: Use

                                         Z = −2(Y − m(X)) · (f(X) − m(X)) + (f(X) − m(X))2

and

                                         E  ((Y − m(X)) · (f(X) − m(X)))2

                                      = E  (f(X) − m(X))2 E  (Y − m(X))2 |X

                                        ≤ L2 E  |f(X) − m(X)| 2 ,

where the last inequality follows from (a).