23. Let u(t ) and v(t ) be the real and imaginary parts, respectively, of the characteristic function of the random variable X. Prove that

(a) E(cos2 t X) = 1 2 [1 + u(2t )],

(b) E(cos sX cos t X) = 1 2 [u(s + t ) + u(s − t )].

Hence, find the variance of cos t X and the covariance of cos t X and cos sX in terms of u and v.

Consider the special case when X is uniformly distributed on [0, 1]. Are the random variables

{cos jπ X : j = 1, 2, . . . } (i) uncorrelated, (ii) independent? Justify your answers. (Oxford 1975F)