1. The problem statement, all variables and given/known data Suppose we have a function, f(x,y) = e^-x * e^-y , 0<=x< ∞, 0<=y<∞, where X and Y are exponential random variables with mean = 1. (For those who may not know, all this means is ∫(x*e^(-x) dx) from 0 to ∞ = 1, and the same for y) Suppose we want to transform f(x,y) into f(z), where the transformation is Z = X-Y. Find f(z) 2. Relevant equations f(x,y) = e^-x * e^-y , 0<=x< ∞, 0<=y<∞ Z = X-Y 3. The attempt at a solution So I decided to transform -Y into W. So we have -Y<=W, which implies Y>=-W ∫e^-y dy from -w to ∞... -e^-y from -w to ∞ 0 + e^-(-w) e^w Differentiate wrt w f(w) = e^w -∞ < w <= 0 So now we have Z = X+W Z-W = X We'll just let W stay as is for this problem. The jacobian of this transformation is 1. So we have f(z) = ∫e^-(z-w)*e^w dw from -∞ to z, This becomes ∫e^-z*e^2w dw from -∞ to z This becomes e^-z * (e^2w)/2 from -∞ to z This becomes e^-z * ((e^2z)-0)/2 Which becomes (e^z)/2 The domain of z is -∞ to ∞, however, this integral does not evaluate to 1. As a matter of fact, it does not even converge. Any help would be greatly appreciated!!