Question

Let ˜x and ˜y be zero-mean, unit variance Gaussian random variables with correlation coefficients, . Suppose we form two new random variables using linear transformations:

Let and be zero-mean, unit variance Gaussian random variables with correlation coefficients, p. Suppose we form two new random variables using linear transformations: Find constraints on the constants a, b, e, and d such that ù and o are inde- pendent.

Answer 1

x^bar and y^bar are zero mean, unit variance random variables with correlation coefficient rho E(x^bar) = E(y^bar) = 0 var (x^bar) = var (y^bar) = 1 Cor (x^bar, y^bar) = cor (x^bar, y^bar)/squareroot var (x^bar) var (y^bar) = rho rightarrow cor (x^bar, y^bar) = rho rightarrow E(x^bar y^bar) - E(x^bar) E(y^bar) = rho rightarrow E(x^bar y^bar) = rho Cor (u^bar, v^bar) = cov (ax^bar + by^bar, cx^bar + dy^bar) = E[(ax^bar + by^bar) (cx^bar + dy^bar)] - E[ax^bar + by^bar] E[cx^bar + dy^bar] = E(ac x^bar^2 + bd y^bar^2 + bc x^bar y^bar + ad x^bar y^bar) - 0 = ac E(x^bar^2) + bd E(y^bar^2) + (ad + bc) E(x^bar y^bar) = (ac + bd) + (ad + bc) rho for u^bar, v^bar to be independent, cor (u^bar, v^bar) = 0 DoubleRightarrow (ac + bd) + (ad + bc) rho = 0 DoubleRightarrow p = -(ac + bd)/(ad + bc)