Problems of the following type, and their solution techniques, were first studied in the 18th century, and the general topic became known as geometric probability.wikipedia
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random setsRandom structure models
Stochastic geometry emphasises the random geometrical objects themselves.
The name appears to have been coined by David Kendall and Klaus Krickeberg while preparing for a June 1969 Oberwolfach workshop, though antecedents for the theory stretch back much further under the name geometric probability.
his needlehis needle problemneedle
(Buffon's needle) What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines?
Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry.
* Wendel's theorem
In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that N points distributed uniformly at random on an n-dimensional hypersphere all lie on the same "half" of the hypersphere.
geometric probability theoryintegral
Integral geometry sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups.
objects derived from random points, and can in part be viewed as a sophisticated branch of multivariate calculus.
Poissonspatial Poisson processinhomogeneous Poisson process
For instance: different models for random lines or for random tessellations of the plane; random sets formed by making points of a spatial Poisson process be (say) centers of discs.
Bertrand's paradoxBertrand paradox
What is the mean length of a random chord of a unit circle? (cf. Bertrand's paradox).
In geometric probability, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle, named after Georges-Louis Leclerc, Comte de Buffon who lived in the 18th century.
Aloha modelspatially distributed telecommunicationsstochastic geometry models
The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.
She is known for her research in discrete geometry, geometric probability, and the theory of random graphs.
At present, several research teams are operating in the Centre like Astrophysics Research Group, Fractional Calculus Research Group, Special Functions Research Group, Statistical Distribution Theory Research Group, Geometrical Probability Research Group, Stochastic Process Research Group, and Discrete Mathematics in Chemistry Research Group.
combinatorialcombinatorial mathematicscombinatorial analysis
Gian-Carlo Rota used the name continuous combinatorics to describe geometric probability, since there are many analogies between counting and measure.