Re: [math-fun] Random variable puzzle
R=-1/sqrt(3) My proof is rather pedestrian, and involves proving that the variance-covariance matrix, with 1 on the diagonal and R off, is positive semi-definite. X can be constructed as U1+R*(U2+U3+U4) (and the others by permutation), where the Ui are independent, zero-mean random variables of equal variance. Cheers, Sébastien On Nov 3, 2015 22:02, "Dan Asimov" <asimov@msri.org> wrote: Suppose X, Y, Z, W are real random variables with a joint distribution such that each one has a finite mean and standard deviation. Suppose that all pairs of these random variables have the same correlation coefficient:* R = rho(X,Y) = rho(X,Z) = rho(X,W) = rho(Y,Z) = rho(Y,W) = rho(Z,W) . Find the minimum possible value of R. —Dan ____________________________________ * The definition of the correlation coefficient rho(U,V) of two random variables U and V is the expected product of their standardizations: rho(U,V) = E( ((U-mu_U)/sigma_U) * ((V-mu_V)/sigma_V)) ) where E is expectation, mu denotes mean and sigma denotes standard deviation. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
You are definitely getting warm! But you are not quite there yet. —Dan
On Nov 4, 2015, at 1:13 PM, Seb Perez-D <sbprzd+mathfun@gmail.com> wrote:
R=-1/sqrt(3)
My proof is rather pedestrian, and involves proving that the variance-covariance matrix, with 1 on the diagonal and R off, is positive semi-definite.
X can be constructed as U1+R*(U2+U3+U4) (and the others by permutation), where the Ui are independent, zero-mean random variables of equal variance.
Cheers,
Sébastien On Nov 3, 2015 22:02, "Dan Asimov" <asimov@msri.org> wrote:
Suppose X, Y, Z, W are real random variables with a joint distribution such that each one has a finite mean and standard deviation.
Suppose that all pairs of these random variables have the same correlation coefficient:*
R = rho(X,Y) = rho(X,Z) = rho(X,W) = rho(Y,Z) = rho(Y,W) = rho(Z,W) .
Find the minimum possible value of R.
—Dan
____________________________________ * The definition of the correlation coefficient rho(U,V) of two random variables U and V is the expected product of their standardizations:
rho(U,V) = E( ((U-mu_U)/sigma_U) * ((V-mu_V)/sigma_V)) )
where E is expectation, mu denotes mean and sigma denotes standard deviation. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
As summary what others have already said and giving a constructive solution for X,Y,Z,W: The covariance matrix reads C(R) = (1-R) I + R * J (where I is the 4x4 identity matrix, and J is the 4x4 all-one matrix) det C(R) >=0 implies R>=-1/3 (not R>= -1/sqrt(3)) If we assume that U1, U2, U3, U4 are independent zero-mean RVs with variance one, we get from the Cholesky decomposition of C(-1/3) (i.e. we search for the matrix L s.t. C(-1/3)=L.L^T) the following solution: X=U1, Y=-1/3 U1 + 2 sqrt(2)/3 U2, Z=-1/3 U1 - sqrt(2)/3 U2 + sqrt(6)/3 U3, W=-1/3 U1 - sqrt(2)/3 U2 - sqrt(6)/3 U3. The coefficient for U4 is always zero. Note, that is not possible to find a symmetric solution (with permutations of indices for U1,U2,U3,U4): If we want to factor C(-1/3) = S^T S, where S=S^T is a symmetric matrix, we obtain C(-1/3)=S^T S = S^2 and C(-1/3)^(1/2) has complex entries. Christoph ________________________________________ From: math-fun [math-fun-bounces@mailman.xmission.com] on behalf of Seb Perez-D [sbprzd+mathfun@gmail.com] Sent: Wednesday, November 04, 2015 10:14 PM To: math-fun Subject: Re: [math-fun] Random variable puzzle R=-1/sqrt(3) My proof is rather pedestrian, and involves proving that the variance-covariance matrix, with 1 on the diagonal and R off, is positive semi-definite. X can be constructed as U1+R*(U2+U3+U4) (and the others by permutation), where the Ui are independent, zero-mean random variables of equal variance. Cheers, Sébastien On Nov 3, 2015 22:02, "Dan Asimov" <asimov@msri.org> wrote: Suppose X, Y, Z, W are real random variables with a joint distribution such that each one has a finite mean and standard deviation. Suppose that all pairs of these random variables have the same correlation coefficient:* R = rho(X,Y) = rho(X,Z) = rho(X,W) = rho(Y,Z) = rho(Y,W) = rho(Z,W) . Find the minimum possible value of R. —Dan ____________________________________ * The definition of the correlation coefficient rho(U,V) of two random variables U and V is the expected product of their standardizations: rho(U,V) = E( ((U-mu_U)/sigma_U) * ((V-mu_V)/sigma_V)) ) where E is expectation, mu denotes mean and sigma denotes standard deviation. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Dan Asimov -
Pacher Christoph -
Seb Perez-D