x,y,z are standard normal random variablesgiven cov(x,z)=0.6,cov(y,z)=0.8,what's the range of cov(x,y) ?I think this should be an exercise problem, but I am rusted.I have tried things like E(x+y+z)^2 >=0, but I can't get sensible results.Another way come in my head is to exploit the definite-positive property of the correlation matrix, this may need the eigenvalue, I think it's too complicated for this problem.please remind me, thank you

thank you, outrun!your solution is very inspired, accutally, the next question of this problem is 'how to simulate the correlated random variables', the common practice is Cholesky decomposition and eigenvalue decomposition.can you clarify this point more clearly?Quote You can see that "b" dosen't show up in the first two constrains Quoteappearently, 'b' indeed shows up in the second constrains.do you mean that whatever b is, the a*d = 0.6 and b*d + c*e =0.8 can be held simultaneously?here's my question:since here x,y,z are standard normals, cov(x,x) = cov(y,y)=cov(z,z) = 1the entire equations should be:a*d = 0.6b*d + c*e =0.8a^2 = 1b^2 + c^2 = 1d^2 + e^2 + f^2 =1a*b = rho_xyassume b = 1,(here rho_xy = cov(x,y) = a*b = 1)then a = 1 or -1 (come from a^2 = 1)c = 0 (come from b^2 + c^2 = 1)d = 0.6 or -0.6 (come from a*d = 0.6)but b*d + c*e =0.8 can never be satisfiedso can I suppose b cannot be equal to 1?

thank you, outrun!your solution is very inspired, accutally, the next question of this problem is 'how to simulate the correlated random variables', the common practice is Cholesky decomposition and eigenvalue decomposition.can you clarify this point more clearly?Quote You can see that "b" dosen't show up in the first two constrains [\q]appearently, 'b' indeed shows up in the second constrains.do you mean that whatever b is, the a*d = 0.6 and b*d + c*e =0.8 can be held simultaneously?here's my question:since here x,y,z are standard normals, cov(x,x) = cov(y,y)=cov(z,z) = 1the entire equations should be:a*d = 0.6b*d + c*e =0.8a^2 = 1b^2 + c^2 = 1d^2 + e^2 + f^2 =1a*b = rho_xyassume b = 1,(here rho_xy = cov(x,y) = a*b = 1)then a = 1 or -1 (come from a^2 = 1)c = 0 (come from b^2 + c^2 = 1)d = 0.6 or -0.6 (come from a*d = 0.6)but b*d + c*e =0.8 can never be satisfiedso can I suppose b cannot be equal to 1?

I'd double-check the problem definition. Any chance that it's supposed to correlation, not covariance? Or is their another constraint (e.g., x,y,z are standard normal random variables with variance 1)?

QuoteOriginally posted by: Traden4AlphaI'd double-check the problem definition. Any chance that it's supposed to correlation, not covariance? Or is their another constraint (e.g., x,y,z are standard normal random variables with variance 1)?yes, x,y,z are standard normal random variables, cov(x,y)=rho_xy

QuoteOriginally posted by: outrunGreat!, good thinking of T$A..now you have at least 3 more values/constraints, the diagonal's of your covariance xx=yy=zz=1, and thus 1. a^2=12. b^2+c^2=13. d^2+e^2+f^2 = 1and the earlier4. a*d = 0.65. b*d+c*e = 0.8so .. can we solvea*b = ?a=1, b=0, c=1, d=0.6, e=0.8, f=0 is a solution.now I have the solution:since the correlation matrix1 rho 0.6rho 1 0.80.6 0.8 1is semi-positive, so its determinant must be no less than 0rho*(0.96-rho)>=0so we get0<rho<0.96

If cov(x,y)=rho_xy, cov(x,z)=rho_xz, cov(y,z)=rho_yz, then:Another approach (that gets the same answer) is:cos(acos(0.6)+acos(0.8)) ≤ rho_xy ≤ cos(acos(0.6)-acos(0.8))0 ≤ rho_xy ≤ 0.96

QuoteOriginally posted by: outrunQuoteOriginally posted by: Traden4AlphaIf cov(x,y)=rho_xy, cov(x,z)=rho_xz, cov(y,z)=rho_yz, then:Another approach (that gets the same answer) is:cos(acos(0.6)+acos(0.8)) ≤ rho_xy ≤ cos(acos(0.6)-acos(0.8))0 ≤ rho_xy ≤ 0.96are you doing that with 3d rotation?I also like the determinant <> 0 approach!It's not "3-D" because the cosine interpretation of rho works for data spaces of any dimension. Rather, the idea is that the angles between the various data vectors must satisfy a kind of triangle inequality. Given rho_xz and rho_yz, which defines the angles between the respective data-vector pairs as acos(rho_xz) and acos(rho_yz), the angle between the X and Y data vectors ( acos(rho_xz ) must lie in the interval between the difference and the sum of the two angles.I, too, like the determinant approach.

Thank you, Traden4Alpha!your solution is very creative!