In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and directional statistics, a wrapped exponential distribution is a wrapped probability distribution

Wrapped distribution

In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n-sphere...

 that results from the "wrapping" of the exponential distribution

Exponential distribution

In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

 around the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

.

Definition

The probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 of the wrapped exponential distribution is


for where is the rate parameter of the unwrapped distribution. This is identical to the truncated distribution

Truncated distribution

In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values...

 obtained by restricting observed values X from the exponential distribution

Exponential distribution

In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

 with rate parameter λ to the range .

Characteristic function

The characteristic function

Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...

 of the wrapped exponential is just the characteristic function of the exponential function evaluated at integer arguments:


which yields an alternate expression for the wrapped exponential PDF:

Circular moments

In terms of the circular variable the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:


where is some interval of length . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:


The mean angle is


and the length of the mean resultant is

Characterisation

The wrapped exponential distribution is the maximum entropy probability distribution

Maximum entropy probability distribution

In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is at least as great as that of all other members of a specified class of distributions....

for distributions restricted to the range for a fixed value of the expectation .