Circular expected value

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Expected value ▷ Circular expected value

Definition: The expected value (or, mean) of a circular random variable $X$ with domain $\mathcal{X} = [0, 2 \pi)$ is defined as the value of $\mu$ satisfying

\[\label{eq:mean-circ-eq} \mathrm{E}\left( e^{iX} \right) = r \cdot e^{i\mu}\]

or, equivalently,

\[\label{eq:mean-circ} \mu = \mathrm{E}(X) = \mathrm{atan2}\left( \mathrm{E}(\sin X), \mathrm{E}(\sin X) \right)\]

where $\sin$ and $\cos$ are the sine and cosine function, respectively, and $\mathrm{atan2}$ is the two-argument arctangent function.

Sources:

Mardia KV, Jupp PE (2000): "Moments and Measures of Location and Dispersion"; in: Directional Statistics, ch. 3.4, pp. 28-31; URL: https://onlinelibrary.wiley.com/doi/book/10.1002/9780470316979; DOI: 10.1002/9780470316979.

Metadata: ID: D230 | shortcut: mean-circ | author: JoramSoch | date: 2026-04-20, 16:01.