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I have two normally distributed random variables. One on the x axis, the other on the y axis, like a complex normal random variable. I'm trying to find the pdf of the angle between a fixed point on the x-y plane(let's say point 1,0) and the vector formed by combining the two random variables.

I tried starting with the pdf of x and y:

p(x,y) =(1/(2πσ^2)) e^(-((x-μ_x)^2+(y-μ_y)^2)/(2σ^2))

where σ is the standard deviation for both random variables

μ_x is the mean for random variable x, and

μ_y is the mean for random variable y

Then, I converted to polar coordinates and integrated over all radii (negative infinity to infinity). Then, it seems like I should integrate from angles 0 to 2pi. However that last integration seems really hard. Does anyone know of another way to solve the pdf of the angle?

In case anyone is curious what the two input random variables actually represent, they are output from an accelerometer

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# Pdf of angle formed by two normal random variables

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