With reference to equations (4.1) and (4.2), let Z1 = U1 and Z2 = −U2 be independent, standard normal variables. Consider the polar coordinates of the point (Z1, Z2), that is, A2 = Z2 1 + Z2 2 and φ = tan−1(Z2/Z1).

(a) Find the joint density of A2 and φ, and from the result, conclude that A2 and φ are independent random variables, where A2 is a chi-squared random variable with 2 df, and φ is uniformly distributed on (−π, π).

(b) Going in reverse from polar coordinates to rectangular coordinates, suppose we assume that A2 and φ are independent random variables, where A2 is chi-squared with 2 df, and φ is uniformly distributed on (−π, π).
With Z1 = A cos(φ) and Z2 = A sin(φ), where A is the positive square root of A2, show that Z1 and Z2 are independent, standard normal random variables.