With reference to equations {4.2) and [4.3), iet 21 = U1 and 22 = U2 be independent, standard normal variables. Consider the polar coordinates ofthe point (21, 22), that is, A2=22+ZZ and go = tan1(22/Z1). 1 2 (a) Find the joint density of A2 and go, and from the resuit, conclude that A2 and go are independent random variables, where A2 is a chi- squared random variable with 2 df, and (p is uniformly distributed on (11, II). (b) Going in reverse from polar coordinates to rectangular coordinates, suppose we assume that A2 and to are independent random variables, where A2 is chi-squared with 2 df, and go is uniformly distributed on {rr, it). With 21 = A cos(qo) and 22 = A sin{qo), where A is the positive square root of A2, show that 21 and 22 are independent, standard normal random variables