Q2 Let (X₁, X₂) be jointly continuous with joint probability density function 2e (²1+2₂) 0 f(x₁, x₂) = 0 <1 <₂<∞ otherwise. Q2(i.) Sketch(Shade) the support of (X₁, X₂). Q2(ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2 (iii.) Let Y₂ = X₂ - X₁. Find the distribution of Y₂ using the distribution function method, i.e., find an expression for Fy₂(y) = P(Y₂ ≤ y) = P(X₂ − X₁ ≤ y) using the joint probability density function (Hint: sketch or shade the region 2 - ₁ ≤ y) and then find the probability density function of Y₂, i.e., fy₂ (y). Q2(iv.) Using the bivariate transformation method, find the joint distribution of Y₁ = 2X₁ and Y₂ = X₂ - X₁. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2(v.) Find the marginal density of Y₂ = X₂ - X₁ and verify that it is the same density function obtained in part Q2 (iii.). Q2 (vi) Find the marginal density of Y₁ = 2X₁. Q2(vii.) Are Y₁ and Y₂ independent?