5.106 Consider two independent standard normal variables whose joint probability density is 1 2 e (z2 1 + z2 2)/2 Under a change to polar

5.106 Consider two independent standard normal variables whose joint probability density is 1 2π e −(z2 1 + z2 2)/2 Under a change to polar coordinates, z1 = r cos( θ ),z2 = rsin( θ ), we have r2 = z2 1 + z2 2 and dz1 dz2 = r dr dθ, so the joint density of r and θ is r e−r2/2 1 2π , 0 <θ< 2π, r > 0 Show that

(a) r and θ are independent and that θ has a uniform distribution on the interval from 0 to 2π;

(b) u1 = θ/2π and u2 = 1−e−r2/2 have independent uniform distributions;

(c) the relations between (u1, u2) and (z1,z2) on page 185 hold [note that 1 − u2 also has a uniform distribution, so ln ( u2 ) can be used in place of ln ( 1 − u2 ) ].