The example 5.44 of the referenced book below concerns about the joint pdf and cdf of two independent Gaussian RV radius $R$ and angle $\theta$ of the point (X,Y). I'd like to know what is the change when the given Gaussian RVs are correlated. here is the explanation when the RVs are independent to be more clear:
Let X and Y be zero mean, unit variance independent Gaussian RV. Given $R$ and $\theta$ respectively as Radius and angle of the point (X,Y):
$R = (X² + Y²)^{1/2}$
$\theta = tan^{-1}(Y/X)$
The joint cdf of $R$ and $\theta$ is:
Figure 1 - joint pdf of $R$ and $\theta$
where:
Figure 2 - Interval of $R$ and $\theta$
That gives us the region of integration of $R(r0,\theta0)$
Figure 3 - Region of integration $R(r0,\theta0)$
When we change variables from cartesian to polar coordinates we obtain:
Figure 4 - Joint cdf $F(r0, \theta0)$
and by taking partial derivates with respect to $r$ and $\theta$ we find the joint pdf:
Figure 5 - Joint pdf $f(r0, \theta0)$
Im having serious troubles looking for how is the behavior of the joint cdf of $R$ and $\theta$ when there is a correlation between X and Y.
Reference:
Leon-Garcia, A. (2008). Probability, Statistics, and Random Processes for Electrical Engineering. Upper Saddle River, NJ: Pearson/Prentice Hall. ISBN: 9780131471221 0131471228