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

If given a known covariance, $$$$cov(X,Y),$$$$ what would the covariance, $$$$cov(RX,RY)$$$$ be, if R is an independant random variable with a variance $$R_v$$ and an expectation $$R_e$$?

I believe the covariance if all variables are mutually independant can be given by: $$$$cov(RX,RY) = R_vX_eY_e$$$$ where $$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?

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 $$c_{xy}=\operatorname{cov}(X,Y)$$.

• Awesome, thank you that answers it perfectly!
– JEK
Commented May 14, 2020 at 18:51
• Actually, is it definitely $E[R^2] = \sigma_r^2 + \mu_r^2$? I was under the impression that is was $E[R^2] = \sigma_r + \mu_r^2$.
– JEK
Commented May 14, 2020 at 19:11
• $\sigma_r^2$ is the variance of $R$ in common notation Commented May 14, 2020 at 19:14
• Oh yeah, sorry I am just learning statistics after doing physics my whole life lol.
– JEK
Commented May 14, 2020 at 19:15