X = [X1 X2]' is a Gaussian (0, CX) vector where
![](https://test.transtutors.com/qimg/1ff7b002-d322-46fe-9a16-9170e9297a03.png)
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
![](https://test.transtutors.com/qimg/4c422420-4ba5-4fd0-8aae-b4463d006ac7.png)
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
![](https://test.transtutors.com/qimg/19c429a4-b932-4412-9348-504cd164916a.png)
![](https://test.transtutors.com/qimg/f0ca45c4-c0e8-4096-a3f7-962fc8867aad.png)
Figure 4.5
![](https://test.transtutors.com/qimg/064a5fd3-0d9f-4e77-b006-ebe1deba35dc.png)
![](https://test.transtutors.com/qimg/b832ea24-2b2b-4630-b689-aff06f699696.png)