Q:

The covariance of two related variables each multiplied by a third independent variable

If given a known covariance,
\begin{equation}
cov(X,Y),
\end{equation}
what would the covariance,
\begin{equation}
cov(RX,RY)
\end{equation}
be, if R is an independant random variable with a variance $R_v$ and an expectation $R_e$?
I believe the covariance if all variables are mutually independant can be given by:
\begin{equation}
cov(RX,RY) = R_vX_eY_e
\end{equation}
where $X_e$ and $Y_e$ are the expectations of X and Y, but is it possible to solve this if there is a covariance between X and Y?

A:

If you also know the means of $X,Y$, you can use the definition of covariance:
$$\begin{align}\operatorname{cov}(RX,RY)&=E[R^2XY]-E[RX]E[RY]\\&=E[R^2]E[XY]-E[R]^2E[X]E[Y]\\&=(\sigma_r^2+\mu_r^2)(c_{xy}+\mu_x\mu_y)-\mu_r^2\mu_x\mu_y\end{align}$$
where $c_{xy}=\operatorname{cov}(X,Y)$.

