Question

Let X₁ and X₂ be jointly continuous random variables withprobability density function f X₁ X₂ . Let Y₁ = X₁ + X₂Y₂= X₁ - X₂ Find the joint density function of Y₁ and Y₂ interms of f X₁, X₂

Answer 1

The joint density function of Y₁ and Y₂ in terms of f(X₁, X₂) is (1/2) * f(X₁, X₂).

Step-by-step explanation:

To find the joint density function of Y₁ and Y₂ in terms of f(X₁, X₂), we need to use the transformation formulas for jointly continuous random variables.

Let's start by finding the Jacobian determinant of the transformation:

|∂(Y₁, Y₂)/∂(X₁, X₂)| = |1 1| = 2

Next, we need to find the joint density function:

f(Y₁, Y₂) = f(X₁, X₂) / |∂(Y₁, Y₂)/∂(X₁, X₂)| = f(X₁, X₂) / 2

Therefore, the joint density function of Y₁ and Y₂ in terms of f(X₁, X₂) is (1/2) * f(X₁, X₂).