Prove that x^-1/x^-1 y^-1 x^-1/x^-1-y^-1=2y^2/y^2-x^2 174454

Question 28 (OR 1st Question) If √(1−𝑥^2 ) √(1−𝑦^2 ) = a (x − y), then prove that 𝑑𝑦/(𝑑𝑥 ) = √(1 − 𝑦^2 )/√(1 − 𝑥^2 ) Finding 𝒅𝒚/𝒅𝒙 would be complicated here To make life easy, we substitute x = sin A y = sin B (As √(1−𝑥^2 )= √(1−sin^2⁡𝐴 )=√(cos^2⁡𝐴 )) And then solve Let's substitute x

Prove that x^-1/x^-1 y^-1 x^-1/x^-1-y^-1=2y^2/y^2-x^2- 1 First make the equation into x^22x1 >= 0 by rearranging and multiplying by x You should find it is (x1)^2 so for any x (not just x>0) this function is greater than or equal to 0 because of ^2 For your situation it is obvious that x>0 which is certainly true for the factorised function Therefore you have proved that x1/x is greaterExample 1 X and Y are jointly continuous with joint pdf f(x,y) = ˆ cx2 xy 3 if 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 0, otherwise (a) Find c (b) Find P(X Y ≥ 1) (c) Find marginal pdf's of X and of Y (d) Are X and Y independent (justify!) (e) Find E(eX cosY) (f) Find cov(X,Y) We start (as always!) by drawing the support set (See below, left) 2 1 2 1 1 x y=1−x y x y support set