If $X_1$ and $X_2$ are unit normal random variables, what about the joint distribution of a transformation.

If $X_1$ and $X_2$ are unit normal random variables, what is the joint probability density of $Y_1 = \frac{X_1 + X_2}{\sqrt{2}}$ and $Y_1 = \frac{X_2 - X_1}{\sqrt{2}}$. Plus, verify if $2X_1X_2$ and $2Y_1Y_2$ have the same distribution.

For the first part I found that $f_{Y_1,Y_2}(y_1,y_2) = \frac{1}{2\pi}e^{-\frac{y_1^2+y_2^2}{2}}$. Looks like the Qui-Square of 1 degree of freedom, but I am not sure, in the books this is valid when $X=Y$, but this is not the case.

Nevertheless, for the second part, seems true (after calculating the marginals of $Y_1$ and $Y_2$), could anyone confirm?

• You did not mention if $X_1$ and $X_2$ are independent or not. Jun 13 '18 at 4:45

Part 1: I think your density is correct, but you have mis-identified it. Hint: maybe check what the joint density of $X_1$ and $X_2$ is?
• Are both the bivariate normal then ? And in that case, Part 2 it is True, because $f_{X_1,X_2}$ and $f_{Y_1,Y_2}$ are identical ! Jun 13 '18 at 0:36