Consider the Cauchy density function

f_X(x) = K / 1 + x² , -∞ ≤ x ≤ ∞

(a) Find K.

(b) Show that var {X} is not finite.

(c) Show that the characteristic function of a Cauchy random variable is M_x(jv) = π Ke^-|v|.

(d) Now consider Z = X₁ + ··· + X_N where the X_i ’s are independent Cauchy random variables. Thus, their characteristic function is 

M_z(jv) = (π K)^N exp (-N|v|)

Show that f_z(z) Cauchy. (f_z(z) is not Gaussian as N → ∞ because var {X_i] is not finite and the conditions of the central-limit theorem are violated.)