use r to find the value of the following

Exercise 17 (#1.46). Let X, and X2 be independent random variables having the standard normal distribution. Obtain the joint Lebesgue density of (Y1, Y2), where Yi = VX7 + X? and Y2 = X1/X2. Are Y, and Yz independent? Note. For this type of problem, we may apply the following result. Let X be a random k-vector with a Lebesgue density fx and let Y = g(X), where g is a Borel function from (R*, B*) to (R*, B* ). Let A1, ...; Am be disjoint sets in B' such that RK - (A] U . . . U Am) has Lebesgue measure 0 and g on A; is one-to-one with a nonvanishing Jacobian, i.e., the determinant Det(Og(x)/0x) # 0 on Aj j =1. .... m. Then Y has the following Lebesgue density: fv(x) = ) Det (Oh; (x)/0r) fx (h;(x)), j=1 where h; is the inverse function of g on Aj, j = 1, .... m.Exercise 1. Let ? be a set, F be o-field on 2, and Ce F. Show that Fo = {CnA : Ac F} is a g-field on C.Exercise 13 (#1.34). Let v be a o-finite measure on a o-field F on , A be another measure with A if v(A) =0 always implies 10 Chapter 1. Probability Theory A(A) = 0, which ensures the existence of the Radon-Nikodym derivative da when v is o-finite (see Shao, 2003, $1.1.2).Exercise 14 (#1.34). Let F, be a o-field on ?;, ; be a o-finite measure on F, and A; be a measure on F, with A,