Geometric probability
Encyclopedia
Problems of the following type, and their solution techniques, were first
studied in the 19th century, and the general topic became known as geometric probability.
For mathematical development see the concise monograph Solomon.
Since the late 20th century the topic has split into two topics with different emphases. Integral geometry
sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups. This topic emphasises systematic development of formulas for calculating expected values associated with the geometric
objects derived from random points, and can in part be viewed as a sophisticated branch of multivariate calculus. Stochastic geometry
emphasises the random geometrical objects themselves. For instance: different models for random lines or for random tessalations of the plane; random sets formed by making points of a spatial Poisson process
be (say) centers of discs.
studied in the 19th century, and the general topic became known as geometric probability.
- (Buffon's needleBuffon's needleIn mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry...
) What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines?
- What is the mean length of a random chord of a unit circle? (cf. Bertrand's paradoxBertrand's paradox (probability)The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités as an example to show that probabilities may not be well defined if the mechanism or method that produces the random variable is not...
).
- What is the chance that three random points in the plane form an acute (rather than obtuse) triangle?
- What is the mean area of the polygonal regions formed when randomly-oriented lines are spread over the plane?
For mathematical development see the concise monograph Solomon.
Since the late 20th century the topic has split into two topics with different emphases. Integral geometry
Integral geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...
sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups. This topic emphasises systematic development of formulas for calculating expected values associated with the geometric
objects derived from random points, and can in part be viewed as a sophisticated branch of multivariate calculus. Stochastic geometry
Stochastic geometry
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns...
emphasises the random geometrical objects themselves. For instance: different models for random lines or for random tessalations of the plane; random sets formed by making points of a spatial Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
be (say) centers of discs.