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

In summary, the distribution of the sum of two circular uniform random variables (RVs) defined on the interval [0, 2π) is analyzed through the concept of circular statistics. The resulting distribution is influenced by the periodic nature of the circular domain, leading to a convolution of circular distributions. The study explores how the sum behaves under various conditions, including the implications of wrapping around the circular range, and provides insights into the characteristics of the resulting distribution, such as its density function and expected values.
  • #1
nikozm
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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!
 
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  • #2
Easiest way is [tex]
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)
[/tex] for [itex]z \in [0, 2\pi)[/itex] and [itex]\Theta[/itex], [itex]\Phi[/itex] are independent and uniformly distributed on [itex][0, 2\pi)[/itex].
 
  • #3
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.
 
  • #4
In the original it is not clear to me that the sum is also mod 2pi. If not then the result will be different.

nikozm said:
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.
 

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

What is the distribution of the sum of two independent circular uniform random variables?

The distribution of the sum of two independent circular uniform random variables, each uniformly distributed over the range [0, 2π), is not uniform. It has a triangular-like shape and is known as the wrapped convolution of two uniform distributions. The resulting distribution tends to have higher density near the middle of the range and lower density near the edges.

How do you derive the probability density function (PDF) for the sum of two circular uniform random variables?

The probability density function (PDF) for the sum of two circular uniform random variables can be derived using the convolution of their individual PDFs. For uniform distributions over [0, 2π), the convolution results in a piecewise linear function that is periodic with period 2π. This function can be expressed using modulo arithmetic and is often referred to as the wrapped triangular distribution.

What are the key properties of the distribution of the sum of two circular uniform random variables?

Key properties of this distribution include its periodicity (with period 2π), its peak around π (the center of the range), and its symmetry about π. The distribution is unimodal and has a mean of π. The variance of the sum of two circular uniform random variables is less than the variance of a single circular uniform random variable.

How does the distribution change if the two circular uniform random variables are not independent?

If the two circular uniform random variables are not independent, the resulting distribution can be significantly different. Dependence can introduce correlations that alter the shape of the distribution. For example, if the variables are positively correlated, the distribution may become more concentrated around certain values, while negative correlation can spread the distribution out more evenly.

Can the sum of two circular uniform random variables be used in practical applications?

Yes, the sum of two circular uniform random variables is used in various practical applications, particularly in fields like signal processing, navigation, and phase synchronization. Understanding this distribution helps in modeling phenomena where angles or phases are involved, such as in the analysis of cyclical data, circular statistics, and in the design of algorithms for circular data processing.

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