I got a problem in exam which is related to Gaussian RV and is as follows: RV 'W' is a gaussian random variable with $N_w(0,1)$ and Y is a random variable defined as:
$$Y = W for |w|<=1 $$ $$Y = -W for |w|>1 $$
It asks me to derive the covariance matrix of W and Y which I know is a matrix consisting of variance and covariance of W and Y \begin{bmatrix} K_y & K_{wy} \\ K_{yw} & K_w \\ \end{bmatrix}
I know that my variance of Y and W are '1' since pdf of Y is also gaussian with variance '1', how can I find K_{wy} as I need to prove whether Y and W are jointly gaussian or not and I need the covariance matrix to prove that. I was trying to derive through following way: Since $cov(w,y)= E{(w-E(w)(y-E(y)}$ I am left with $E(WY)$ which is for:
E(WY) = E(y^2) for |y|<=1 $$-$$ E(wy) = E(-y^2) for |y|>1
The first is the variance which is '1' but second gives me variance '-1' which is wrong, can someone help me as to what I am doing wrong, thanks