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Suppose there are two 2-d multivariate normal distributions , as in the image below: Multivariate normal distributions

There is no correlation between the x and y components in the distributions, and the variance for each dimension is the same.

Is it correct to say that the angle between the two distributions could be expressed as a wrapped normal distribution, with $\mu _{\theta } = atan2(\mu_{2_y} - \mu_{1_y}, \mu_{2_x} - \mu_{1_x})$?

If so, I am not sure how to derive the variance of this wrapped distribution, based on the two multivariate normal distributions. Intuitively, the variance of the wrapped distribution should decrease as the distance between $\mu_1$ and $\mu_2$ increases. Also, the variance should increase as $\sigma^2_1$ and $\sigma^2_2$ increase.

How can I derive the variance and the standard deviation for the wrapped distribution of the angle between the two distributions?

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  • $\begingroup$ Is it that, for each observation, you sample one point from the blue density and a second from the green density and then you compute the angle between the x-axis and the line that connects your two points? And you want to know the distribution of this angle? $\endgroup$
    – frank
    Sep 2, 2022 at 7:09
  • $\begingroup$ @frank yes, I think that would effectively give the answer that I want. $\endgroup$
    – r0gi
    Sep 5, 2022 at 0:00

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