# Covariance matrix under multiplication with independent random variable.

assume two dependent random variables X and Y with their covariance matrix $$\Sigma_{ij}$$. Now assume one multiplies X with another independent random variable Z. The variance of the resulting random variable of the product is then given by (correct me if I am wrong):

$$Var[X*Z] = (E[X])^2Var[Z] + (E[Z])^2Var[X] + Var[X]Var[Z]$$

My questions is: How does this multiplication influence the covariance matrix $$\Sigma_{ij}$$? Intuitively I would say it just influences the element $$\Sigma_{XX}$$ on the main diagonal of $$\Sigma$$ (replaced by $$Var[X*Z]$$). But I am not quite sure how to show that.

Thanks for yours help :-)

Best

• If $Z$ was an an almost surely constant $z$ then the covariance of $XZ$ with $Y$ would be $z$ times the covariance of $X$ with $Y$ – Henry Feb 6 at 15:15
• Several terms are missing in your formula. Related questions have been addressed and solved by George W. Bohrnstedt and Arthur S. Goldberger, 1969, "On the Exact Covariance of Products of Random Variables," Journal of the American Statistical Association, Vol. 64, No. 328, 1439-1442. – Bertrand Feb 6 at 15:39