Question

Suppose ( (X₁, X₂) ) have the joint pdf ( f(x₁, x₂) = 2e^{-(1+2)} ), ( 0 < X₁ < X₂ < infty ). Find the pdf of ( Y₁ = X₁ - X₂ ) using the change of variable technique.

Answer 1

To find the pdf of Y₁ = X₁ - X₂ using the change of variable technique, you need to determine the joint cumulative distribution function (CDF) of X₁ and X₂ and then use the change of variable technique to express the joint CDF in terms of Y₁. Differentiate the joint CDF to find the pdf of Y₁.

Step-by-step explanation:

To find the pdf of Y₁ = X₁ - X₂ using the change of variable technique, we first need to determine the joint cumulative distribution function (CDF) of X₁ and X₂. Since the joint pdf of X₁ and X₂ is given as f(x₁, x₂) = 2e^{-(1+2)}, we can integrate this pdf over the appropriate range to obtain the joint CDF. In this case, the range is 0 < X₁ < X₂ < ∞.

Next, we use the change of variable technique to express the joint CDF in terms of Y₁ by substituting X₁ = Y₁ + X₂. Now, we can differentiate the joint CDF to find the pdf of Y₁.

By following these steps, we can determine the pdf of Y₁ = X₁ - X₂.