Given $X_1,X_2$ two random variables, both $X$'s are independent, normal random variables,
(a) How can I find the joint distribution of $\frac{X_1−X_2}{\sqrt{2}}$ and $\frac{X_1+X_2}{\sqrt{2}}$
For (a) I know that i can use transformations:
$X_1,X_2$~$N(0,1)$
$Y_1=\frac{X1+X2}{\sqrt{2}}$ and $Y_2=\frac{X1+X2}{\sqrt{2}}$ $-\infty<y_1,y_2<\infty$
$X_1=\frac{\sqrt{2}Y_1+Y_2}{2}$ and $X_1=\frac{\sqrt{2}Y_1-Y_2}{2}$
$f(y_1,y_2)=f(w_1(y_1,y_2),w_2(y_1,y_2)|J|$
Can anyone tell me if I'm in the right direction and what is |J| (I know is the Jacobian, but what values go inside the matrix?)
Best Answer
Let $y_1=\frac{x_1-x_2}{\sqrt{2}}, and\ y_2=\frac{x_1+x_2}{\sqrt{2}}$, then $x_1^2+x_2^2=y_1^2+y_2^2$. The absolute value of the Jacobian of the transformation =1, so $ dx_1dx_2=dy_1dy_2$. As a result the joint distribution function for $(Y_1,Y_2)$ is $P(Y_1\lt u_1,Y_2\lt u_2)=\int_{-\infty}^{u_1}\int_{-\infty}^{u_2}\frac{1}{\sqrt{2\pi}}e^{-(y_1^2+y_2^2)}dy_2dy_1$, which shows that $Y_1$ and $Y_2$ are independent.
To answer your question the Jacobian matrix is $\bigg(\frac{\frac{\partial{x_1}}{\partial{y_1}}}{\frac{\partial{x_1}}{\partial{y_2}}}\frac{\frac{\partial{x_2}}{\partial{y_1}}}{\frac{\partial{x_2}}{\partial{y_2}}}\bigg)$