In probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a circular distribution or polar distribution is a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 of a random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

 whose values are angles, usually taken to be in the range . A circular distribution is often a continuous probability distribution, and hence has a probability density

Probability density

Probability density may refer to:* Probability density function in probability theory* The product of the probability amplitude with its complex conjugate in quantum mechanics...

, but such distributions can also be discrete, in which case they are called circular lattice distributions. Circular distributions can be used even when the variables concerned are not explicitly angles: the main consideration is that there is not usually any real distinction between events occurring at the lower or upper end of the range, and the division of the range could notionally be made at any point.

Graphical representation

If a circular distribution has a density


it can be graphically represented as a closed curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

where the radius is set equal to

and where a and b are chosen on the basis of appearance.

Example

By computing the probability distribution of angles along a handwritten ink trace,
a lobe-shaped polar distribution emerges. The main direction of the lobe in the
first quadrant corresponds to the slant

Slant (handwriting)

Slant is the predominant angle of the downward stroke in Western handwriting. A good basis for its estimation is the point of the handwritten curve where the velocity has its peak value in the...

 of handwriting (see: graphonomics

Graphonomics

Graphonomics is the interdisciplinary field directed towards the scientific analysis of the handwriting process and the handwritten product. Researchers in handwriting recognition, forensic handwriting examination, kinesiology, psychology, computer science, artificial intelligence, paleography and...

).

An example of a circular lattice distribution would be the probability of being born in a given month of the year, with each calendar month being thought of as arranged round a circle, so that "January" is next to "December".

See also

• Circular mean
• Circular uniform distribution
  Circular uniform distribution
  In probability theory and directional statistics, a circular uniform distribution is a probability distribution on the unit circle whose density is uniform for all angles.- Description :The pdf of the circular uniform distribution is:...
  
  
• Directional statistics#Circular and higher dimensional distributions