Given
$$ Y_1 = X_1 – X_2 , Y_2 = X_1 + X_2, $$
how can I find the joint probability density function of $$(Y_1,Y_2)?$$
The $X$'s are independent normal random variables and both $Y$'s are random variables as well.
[Math] Finding the joint probability density function of two random variables
probability distributionsrandom variables
Best Answer
If I were you, I would use transformations.
$$ Y1 = X1 - X2 = u_1(x_1,x_2)$$ $$ Y2 = X1 + X2 = u_2(x_1,x_2)$$ $$ X_1 = \frac{Y_1+Y_2}{2}=w_1(y_1,y_2)$$ $$ X_2 = \frac{Y_1-Y_2}{2}=w_2(y_1,y_2)$$
$$ f(y_1,y_2) = f(w_1(y1, y2),w_2(y1,y2)|J|$$
j is jaconian
I solved same problem with standard normal distributions. I added picture of my process. or You can solve this problem by using distribution method.
good luck :)