Two random variables X1, X2-M0,1) are correlated with correlation coefficient P[X1, X2] = E[X1, X2] =...

  

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Two random variables X1, X2-M0,1) are correlated with correlation coefficient P[X1, X2] = E[X1, X2] = p. State the vector of mean values and the covariance matrix for the vector X = (X1, X2). %3D A computation of the eigenvalues yields 1+p and 1 - p and the eigenvectors are (-1,1)"/V2 and (1,1)/V2. Write down the discrete Karhunen-Loève expansion. Check by a direct computation that (X1, X2) have the correct mean value and variance. Hint: for uncorrelated random variables U,V there holds V[aU + bV] = a²V[U]+b?V[V]. Explain what happens if p → 1 and p+ -1. Two random variables X1, X2-M0,1) are correlated with correlation coefficient P[X1, X2] = E[X1, X2] = p. State the vector of mean values and the covariance matrix for the vector X = (X1, X2). %3D A computation of the eigenvalues yields 1+p and 1 - p and the eigenvectors are (-1,1)"/V2 and (1,1)/V2. Write down the discrete Karhunen-Loève expansion. Check by a direct computation that (X1, X2) have the correct mean value and variance. Hint: for uncorrelated random variables U,V there holds V[aU + bV] = a²V[U]+b?V[V]. Explain what happens if p → 1 and p+ -1.

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For p tends to 1 we have the eigenvalue to be at max... View the full answer