Assume that we have $X_1,X_2,X_3$ that are jointly Gaussian, $E(X_i)=0$ and with non-zero covariances. We also have:
$$Y_1=X_1$$
$$Y_2=X_2−X_1$$
$$Y_3=X_3−X_2$$
With $0$ covariances among all $Y_i$ and $E(Y_i^2)=1$
Is the following operation valid:
$$E(Y_3^2|X_1,X_2)=E(Y_3^2|Y_1,Y_2)=E(Y_3^2)=1$$
Also consider: $$E(Y_3^2|X_1,X_2)= E(X_3^2|X_1,X_2)-2E(X_3|X_1,X_2)X_2-X_2^2$$
Which should be impossible to solve with the current information.
I know that in the case of jointly gaussian variables a correlation of $0$ implies independence and all $X_i$ are jointly gaussian but does this property extend to all the $Y_i$?
Best Answer
Yes, your argument is correct. An affine (in this case: linear) transformation of jointly Gaussian variables again yields jointly Gaussian variables; see e.g. Wikipedia. Since fixing $X_1$ and $X_2$ is equivalent to fixing $Y_1$ and $Y_2$, the first equality holds, and since the $Y_i$ are jointly normal with zero covariance, the second equality holds.