Editing Probability distribution (section)

=== Some specialized applications of probability distributions ===

* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by <math display="inline">P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.<ref>{{Cite book| title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|others=Thoman, John W., Jr., 1960-| year=2014| isbn=978-1-68015-835-9 |location=[Mill Valley, California]|pages=403–406|oclc=927509011}}</ref>
* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provides the power flow calculation also in term of probability distribution.<ref>{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.| last2=Chen|first2=Z.| last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review| doi=10.1109/drpt.2008.4523658|s2cid=18669309}}</ref>
* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.<ref>{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity | first = Rajib| isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}</ref>