Fraction of a Circle Covered by Arcs of a Given Length

This Demonstration shows the probability density function for the fraction of a circle with unit circumference covered by random arcs, each of length . This probability distribution is a mixed distribution. The discrete probability mass is shown and captioned in red, and the continuous probability density is shown in blue (scaled by ). and are the mean and variance of the fraction covered.

The probability density function is continuous between 0 and , where is the minimum of 1 and , and has a discrete probability at . For , the discrete part is the probability that all arcs do not overlap. When , the discrete part is the probability of complete coverage of the circle. The probability density function and probability mass functions shown are calculated in [1].

Reference

[1] H. Solomon, Geometric Probability, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1978 pp. 75–96.