Strategic Practice and Homework 8 - Projects at Harvard

(a) Explain intuitively why R and ✓ are independent. Then prove this by finding thejoint PDF of (R, ✓).Intuitively, this makes sense since the joint PDF of X, Y at a point (x, y) onlydepends on the distance from (x, y) totheorigin,notontheangle,soknowingRgives no information about ✓. The absolute Jacobian is r (as shown on the mathreview handout), sof R,✓ (r, t) =f X,Y (x, y)r = r · g(r 2 )for all r 0,t 2 [0, 2⇡). This factors as a function of r times a (constant) functionof t, soR and ✓ are independent with ✓ ⇠ Unif(0, 2⇡).(b) What is the joint PDF of (R, ✓) when(X, Y ) is Uniform in the unit disc {(x, y) :x 2 + y 2 apple 1}?We have f X,Y (x, y) = 1 for ⇡ x2 + y 2 apple 1, so f R,✓ (r, t) = r for 0 apple r apple 1,t 2 [0, 2⇡)⇡(and the PDF is 0 otherwise). This says that R and ✓ are independent with marginalPDFs f R (r) =2r for 0 apple r apple 1andf ✓ (t) = 1 for 0 apple t