I want to solve this problem by jacobian transformation Not

I want to solve this problem by jacobian transformation. Not by using CDFs

8. Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, X1 and X2, of the respective components are independent and identically distributed with an exponential distribution with Theta = 2. The cost of operating the device is Z = 2Y1 + Y2, where Y1 = min (X1, X2) and Y2 = max (X1, X2). Compute E(Z). I want to solve this problem by jacobian transformation. Not by using CDFs

Solution

x1 = 2y1

x2 = y2

Then J = 2 0

0 1

= 2

Hence pdf of Z = 2y1+y2 is

= Exponential with parameter 4+2 =6

Mean = 6