Question

Let each of the independent random variables X1 and X2 have the density function f(x) - e^-x for 0<x< inf., and f(x) = 0, otherwise. What is the joint density of Y1 = X1 and Y2 = 2X1 + 3X2 and the domain on which this density is positive?

Answer 1

The joint density of X1 and X2 is,

for x1 > 0, x2 > 0

Given,

Y1 = X1 and Y2 = 2X1 + 3X2

The unique solution of X1 and X2 are,

X1 = Y1 and X2 = (Y2 - 2Y1) /3

The Jacobian is,

The domain on which this density is positive are,

x1 > 0 => y1 > 0

x2 > 0 =>  (y2 - 2y1) /3 > 0 => y2 > 2y1 => y2 / 2 > y1

Thus,

for 0 < y1 < y2/2 <