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X = [X1 X2]' is a Gaussian (0, CX) vector where
Thus, depending on the value of the correlation coefficient ρ, the joint PDF of X1 and X2 may resemble one of the graphs of Figure 4.5 with X1 = X and X2 = Y. Show that X = QY where Q is the θ = 45? rotation matrix (see Problem 5.7.6) and Y is a Gaussian (0, CY) vector such that This result verifies, for ρ ≠ 0, that the the PDF of X1 and X2 shown in Figure 4.5, is the joint PDF of two independent Gaussian random variables (with variances 1 + ρ and 1 - ρ) rotated by 45 degrees.
Problem 5.7.6
The 2 × 2 matrix
Is called a rotation matrix because y = Qx is the rotation of x by the angle θ. Suppose X = [X1 X2] is a Gaussian (0, CX) vector where
Figure 4.5
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