If $X_1$ and $X_2$ are jointly Gaussian with zero mean and Covariance Matrix
$$\begin{bmatrix}
\sigma_1^2&\sigma_1\sigma_2\\
\sigma_1\sigma_2&\sigma_2^2\\
\end{bmatrix}
$$
Define $Y_1$ and $Y_2$ as
$$Y_1 = \dfrac{X_1}{\sigma_1} + \dfrac{X_2}{\sigma_2}$$ and $$Y_2 = \dfrac{X_1}{\sigma_1} – \dfrac{X_2}{\sigma_2}$$
The question is to find if $Y_1$ and $Y_2$ are uncorrelated and/or independent. Since $E[Y_1 Y_2] = 0 = E[Y_1]E[Y_2]$, they are uncorrelated.
But for $Y_2$, both the mean and variance is coming to be zero. But, my intuition about $Y_2$ says it cannot be zero all the time.
Can anyone help where I am thinking wrong?
Best Answer
Indeed $EY_2^{2}=0$ so $Y_2$ is the zero random variable! This implies in particular that $Y_1$ and $Y_2$ are independent.