Thus depending on the value of the correlation coefficient

Question

X = [X₁ X₂]' is a Gaussian (0, C_X) 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, C_X) vector where

Figure 4.5