# Covariance of two variables that are products of shared random variables

How to analytically express cov(X,Y), when:

X=C*A/(A+B)


and

Y=C*B/(A+B)


Here C, A and B are independent variables with normal distributions.

More specifically I would like to express cov(X,Y) using the expected means of each of the input variables C, A and B, as well as their standard deviations or variances.

A is a normal distribution with an expected mean E(A) and a variance var(A)
B is a normal distribution with an expected mean E(B) and a variance var(B)
C is a normal distribution with an expected mean E(C) and a variance var(C)


Thank you so much in advance!

• if this is HW please add the self-study tag – Antoine Jun 9 '16 at 9:42
• Thank you for asking but it is for my research at work. I have really searched a lot the last few days for the answer, but I can only find examples where X and Y are linear combinations of random variables like X=aA+bB+cC (with small a, b and c as constants) but I cannot find solutions where X is a function that is the product of a combination of A, B and C. – Astrid Marie Jun 9 '16 at 10:18
• I think that would help if you could show some of those examples you refer to and then explain how it is different in your case. – Antoine Jun 9 '16 at 10:27
• Thanks for the tip. The question ressembles a bit this stats.stackexchange.com/questions/137571/covariance-of-product , but the example in the link is in matrix notation and it is not clear for me if Xi and Xj in the example are correlated, so I cannot used this directly. – Astrid Marie Jun 9 '16 at 10:36
• Given that this involves the ratio of Normals, convergence may be an issue – wolfies Jun 9 '16 at 11:24

One can find a general solution, without assuming Normality. In particular, if $A$, $B$ and $C$ are independent, and noting that the Covariance operator is the {1,1} central moment, then $\text{Cov}( \frac{C A}{A+B},\frac{C B}{A+B})$ is:

where I am using a developmental version of the CentralMomentToCentral function in mathStatica (alas, not in any public release yet).

The reason the solution does not converge (assuming Normal parents) is because $A+B$ is Normal, and thus $E[\frac{1}{A+B}]$ is the expectation of the inverse of a Normal variable, which is known not to converge (see, for instance, Mean and variance of inverse of a normal RV)

• Thank you so much Wolfies! It is extremely helpful to see your answer, and I really appreciate that you took your time to work with this problem. Now I try to implement the result in my calculations and compare with the simulations :-) The mathStatica seems like a really powerful tool. – Astrid Marie Jun 10 '16 at 12:58
• Just one little question, I see that the C in your code has a different colour than A and B. Is there any reason for this? – Astrid Marie Jun 10 '16 at 13:21