Let X be a random variable and t be a real number such that the moment E[(X − t)2]

exists.

(i) Verify that E[(X − t)2] = E(X2) − 2tE(X) + t2.

(ii) Show that E[(X − t)2] = Var(X) + (???? − t)2.

(iii) Use the result in (ii) to establish that the minimum value of the function h(t) = E[(X − t)2]

is achieved at t = ????, and that min t∈ℝ

E[(X − t)2] = Var(X).