Editing Probability distribution (section)

==Other kinds of distributions==
[[File:Rabinovich_Fabrikant_2314.png|right|thumb|300px|Figure 8: One solution for the [[Rabinovich–Fabrikant equations]]. What is the probability of observing a state on a certain place of the support (i.e., the red subset)?]]
Absolutely continuous and discrete distributions with support on <math>\mathbb{R}^k</math> or <math>\mathbb{N}^k</math> are extremely useful to model a myriad of phenomena,<ref name='ross' /><ref name='dekking' /> since most practical distributions are supported on relatively simple subsets, such as [[hypercubes]] or [[ball (mathematics)|balls]]. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves <math>\gamma: [a, b] \rightarrow \mathbb{R}^n</math> within some space <math>\mathbb{R}^n</math> or similar. In these cases, the probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it.<ref name='alligood'>{{cite book|author1=Alligood, K.T.|author2=Sauer, T.D.|author3=Yorke, J.A.|year=1996|title=Chaos: an introduction to dynamical systems|publisher=Springer}}</ref>

One example is shown in the figure to the right, which displays the evolution of a [[system of differential equations]] (commonly known as the [[Rabinovich–Fabrikant equations]]) that can be used to model the behaviour of [[Langmuir waves]] in [[plasma (physics)|plasma]].<ref>{{cite journal|author1=Rabinovich, M.I.|author2=Fabrikant, A.L.|year=1979|title=Stochastic self-modulation of waves in nonequilibrium media|journal=J. Exp. Theor. Phys.|volume=77|pages=617–629|bibcode=1979JETP...50..311R}}</ref> When this phenomenon is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system.<ref>Section 1.9 of {{cite book|author1=Ross, S.M.|author2=Peköz, E.A.|year=2007|title=A second course in probability|url=http://people.bu.edu/pekoz/A_Second_Course_in_Probability-Ross-Pekoz.pdf}}</ref><ref name='alligood' />

This kind of complicated support appears quite frequently in [[dynamical systems]]. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let <math>t_1 \ll t_2 \ll t_3</math> be instants in time and <math>O</math> a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set <math>O</math> would be equal in interval <math>[t_1,t_2]</math> and <math>[t_2,t_3]</math>, which might not happen; for example, it could oscillate similar to a sine, <math>\sin(t)</math>, whose limit when <math>t \rightarrow \infty</math> does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future.<ref>{{cite book|last=Walters|first=Peter|title=An Introduction to Ergodic Theory|year=2000|publisher=Springer}}</ref> The branch of dynamical systems that studies the existence of a probability measure is [[ergodic theory]].

Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely continuous" or "discrete" depending on whether the support is uncountable or countable, respectively.