I am trying to solve a problem where I need to find Covariance between X and Y random variables. I am given that $Cov(X) = a$ and $$Cov(Y) = \left(\begin{array}{ccc}
3&-2\\
-2&5
\end{array}\right)$$
But I am not sure about how can I find Covariance between 2*2 and 1*1 matrix. Can someone please help me or point out to some online resource for same?
I tried searching on google a lot but it didn't help.
Best Answer
Given the information provided - using only the variance of X (recall that if X is 1x1, you can only compute the variance $Var(x)=Cov(X,X)=a$) and the covariance of Y - it is impossible to calculate the covariance between X and Y. You must make a further assumption to tackle the issue. Before that, we should be more specific and write Y=$(Y_1,Y_2)$ where $Y_i$ is the i-th random variable of the vector Y. The following assumption need to me made:
You must assume a common density between X and $Y_i$ or that the distributions are independent. This follows by the definition of the covariance matrix of a random vector $V=(X,Y_1,Y_2)$: https://en.wikipedia.org/wiki/Covariance_matrix
That's because each entry of this 3x3 covariance matrix consists out of a 1x1 covariance. So you must assume that $Cov(X,Y_i)=E(XY_i)-E(X)E(Y_i)$ exists, which implies that the common density $P(X,Y_i)$ must exist or that the two random variables are independent distributed.