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I have four random variables, A, B, C, D, with known mean and variance. As well:

  • Cov(A,B) is known and non-zero
  • Cov(C,D) is known and non-zero
  • A and C are independent
  • A and D are independent
  • B and C are independent
  • B and D are independent

I then create two new random variables:

  • X = A*C
  • Y = B*D

Is there any way to determine Cov(X,Y) or Var(X+Y)?

If not exactly, is there any way to estimate it? What if I could determine the distributions of A and B and C and D?

kjetil b halvorsen
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Albeit
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2 Answers2

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If I did this correctly:

\begin{eqnarray} \text{Cov}(AC,BD) &=&E(ABCD) - E(AC)E(BD)\\ &=&E(AB)E(CD) - E(A)E(C)E(B)E(D)\\ &=&[E(AB)-E(A)E(B)][E(CD)-E(C)E(D)]+E(A)E(B)[E(CD)-E(C)E(D)]+E(C)E(D)[E(AB)-E(A)E(B)]\\ &=&\text{Cov}(A,B)\text{Cov}(C,D)+E(A)E(B)\text{Cov}(C,D)+E(C)E(D)\text{Cov}(A,B)\end{eqnarray}

Glen_b
  • 282,281
  • Note: The E(ABCD)-E(AC)E(BD) = E(AB)E(CD) - E(A)E(B)E(C)E(D)is an approximation in small samples. (In small samples, the sample independence may not be the population independence.) – ivo Welch Jul 28 '20 at 21:31
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    There's no sample-level approximation here at all. All the calculations are in relation to random variables. – Glen_b Oct 11 '22 at 22:27
5

A result in Bohrnstedt, George W., and Arthur S. Goldberger. “On the Exact Covariance of Products of Random Variables.” Journal of the American Statistical Association, vol. 64, no. 328, 1969, pp. 1439–42. JSTOR, https://doi.org/10.2307/2286081. Accessed 22 Jan. 2024. provides the general formulae. Assuming multivariate normality it is:

$$ \mathrm{cov}(xy, uv) = \mathrm{E}(x)\,\mathrm{E}(u)\, \mathrm{cov}(y, v) + \mathrm{E}(x)\,\mathrm{E}(v)\,\mathrm{cov}(y, u) + \\ \mathrm{E}(y)\,\mathrm{E}(u)\, \mathrm{cov}(x, v) + \mathrm{E}(y)\,\mathrm{E}(v)\,\mathrm{cov}(x, u) + \\ \mathrm{cov}(x, u)\, \mathrm{cov}(y, v) + \mathrm{cov}(x, v)\,\mathrm{cov}(y, u) $$

In your case: x = A, y=C, u=B, v =D and the only non cero covariances are cov(A,B)= cov(x,u) and cov(C,D)=cov(y,v) so:

$$ \mathrm{cov}(AC, BD) = \mathrm{E}(A)\,\mathrm{E}(B)\, \mathrm{cov}(C, D) + \\ \mathrm{E}(C)\,\mathrm{E}(D)\,\mathrm{cov}(A, B) + \\ \mathrm{cov}(A, B)\, \mathrm{cov}(C, D) $$

Which is the result of the previous post.

Sycorax
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mcs
  • 51
  • please provide full reference for your link in case it dies in the future. Thanks! – Antoine Jul 11 '20 at 13:59
  • Title: On the Exact Covariance of Products of Random Variables Author: George W. Bohrnstedt and Arthur S. Goldberger. Year= 1969. Journal= JASA – ivo Welch Jul 28 '20 at 20:56
  • So this only works for Gaussian variables? – a06e Jul 14 '23 at 17:42
  • The JSTOR link is behind a paywall. But there's a free version here that looks like it's probably the same document. https://pj.freefaculty.org/guides/crmda_workshops/sem/Archive/sem-4/literature/Bohrnstedt_Goldberger_1969.pdf – Stephen Jewson Jan 22 '24 at 20:03