6.13 In this problem we shall derive the characteristic function of a normal using a differential equation approach.

a) Let f(x) = e− x2 2 . Show that the function f satisfies d

dxf + xf = 0.

b) Show that if Z ∼ N(0, 1) then its characteristic function ϕZ satisfies the same equation.

c) Deduce that

ϕZ(t) = Ce− t2 2 .

d) Compute the constant C from the relation