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I have a continuous random variable over a circle, so that the probability density function $\rho(\theta) = \rho(\theta+2\pi)$. I should be able to calculate its expected value and standard deviation, although I may need to define them slightly differently in order to achieve the same intuitive goals. The properties I am looking for are:

  1. The circle statistics should reduce to regular flat-space statistics whenever my PDF is highly concentrated in one part of the circle (because curves locally appear like lines).

  2. They should transform sensibly under a coordinate shift $\rho(\theta) \rightarrow \rho'(\theta) =\rho(\theta+\phi)$, carrying $E[\rho]\rightarrow E[\rho']=E[\rho]+\phi$ and Var$[\rho]\rightarrow $Var$[\rho']=$Var$[\rho]$. We desire these properties because the angle is an arbitrary coordinate and choice of coordinate system shouldn't determine anything significant about our answers.

How should the expected value and variance be defined so that they satisfy these properties?

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As suggested by Jim B, The $n$th moment of a circular distribution is defined as, $$ m_n = \int_\phi^{\phi+2\pi} \rho(\theta) e^{in\theta}\, \mathrm{d\theta}, $$ where $\phi$ is any angle and the probability density $\rho$ is periodic with period $2\pi$. It is illuminating to see how these moments vary with coordinate changes. Rotating $\rho$ by an angle $\varphi$ we have, $$ m_n' = \int_\phi^{\phi+2\pi} \rho(\theta-\varphi) e^{in\theta}\, \mathrm{d\theta} = e^{in\varphi}\int_{\phi-\varphi}^{\phi-\varphi+2\pi} \rho(\theta') e^{in\theta'}\, \mathrm{d\theta'} = e^{in\varphi}m_n $$ Where in the last step we used the fact that periodic functions have the same integral over any range that is exactly as long as their period. It is clear that $m_0$ is real and equal to 1 for well-behaved distributions. $m_1$ has a complex argument equal to the average angle, and by $e^{i \varphi}$ transforms as we would expect with rotating coordinate systems. For $n>1$, we can conclude that the complex argument of $m_{n>1}$ cannot be meaningful as a property of $\rho$, as it is not invariant with respect to rotations.

To see how we might recover the "usual" moments from the circular moments, suppose that $\rho(\theta)$ is concentrated in a small region that we may without loss put near $\theta=0$. Expanding $e^{in\theta}$ around that point, we arrive at, $$ m_n \approx \sum^\infty_{j=0}\frac{(in)^j}{j!}\int_{-\pi}^{\pi} \theta^j \rho(\theta) \, \mathrm{d\theta} $$ It is clear that the "usual" moments are all hidden together within $m_n$, summed with weights that depend on $n$. They may be isolated with polynomials of $m_n$ for many $n$, although this is not usually done.

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