62. Let X1 and X2 be independent, each having a standard normal distribution. The pair (X1, X2) corresponds to a point in a two-dimensional coordinate system. Consider now changing to polar coordinates via the transformation Y2  g arctan a X2 X1 b X1 0, X2  0 arctan a X2 X1 b  2p X1 0, X2  0 arctan a X2 X1 b  p X1  0 0 X1  0 Y1  X21

 X22

 e 8x1x2x3 0  x1  1, 0  x2  1, 0  x3  1 0 otherwise f1x1, x2, x3 2 from which .

Obtain the joint pdf of the new variables and then the marginal distribution of each one. Note: It would be nice if we could simply let Y2  arctan(X2/X1), but in order to ensure invertibility of the arctan function, it is de ned to take on values only between

p/2 and p/2. Our speci cation of Y2 allows it to assume any value between 0 and 2p.