Distribution of sum of two circular uniform RVs in the range [0, 2 pi)

[nikozm]

TL;DR Summary

distribution; uniform

Hello,

I would like to know the analytical steps of deriving the distribution of sum of two circular (modulo 2 pi) uniform RVs in the range [0, 2 pi).

Any help would be useful

Thanks in advance!

 

[pasmith]

Easiest way is $$P(0 \leq Z = (\Theta + \Phi) \mod 2 \pi < z ) = P(0 \leq \Theta + \Phi < z) + P(2\pi \leq \Theta + \Phi < z + 2\pi)$$ for $z \in [0, 2\pi)$ and $\Theta$, $\Phi$ are independent and uniformly distributed on $[0, 2\pi)$.

 

[nikozm]

I try to utilize this formula to a similar case, but the result seems too complicated. What if one of two RVs is a circular (mod 2 pi) uniformly distributed in [0, 2 pi) and the other one is an independent uniform RV in the range [-2^(-q) pi, 2^(-q) pi], where q is a nonnegative integer greater or equal than one. I presume that their sum is also a uniform RV, but I am not sure about its range.

Can you help me on this.

Thank you so much in advance.

 

[Hornbein]

In the original it is not clear to me that the sum is also mod 2pi. If not then the result will be different.

I try to utilize this formula to a similar case, but the result seems too complicated. What if one of two RVs is a circular (mod 2 pi) uniformly distributed in [0, 2 pi) and the other one is an independent uniform RV in the range [-2^(-q) pi, 2^(-q) pi], where q is a nonnegative integer greater or equal than one. I presume that their sum is also a uniform RV, but I am not sure about its range.

Can you help me on this.

Thank you so much in advance.

Same here. Is the result mod 2pi? Unfortunately Wikipedia gives two definitions of the mod operator and the answer differs in the two cases. So you are right to be uncertain.

Wikipedia : In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another.