The fundamental importance of differential equations with three singularities in Mathematical Statistics
Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie/South African Journal of Science and Technology
| Field | Value | |
| Title | The fundamental importance of differential equations with three singularities in Mathematical Statistics Die fundamentele belang van differensiaalvergelykings met drie singulariteite vir die Wiskundige Statistiek | |
| Creator | Steyn, H. S. | |
| Description | It is well-known that the solution of a second order linear differential equation with at most five singularities plays a fundamental role in Mathematical Physics. In this paper it is shown that this statement also applies to Mathematical Statistics but with the difference that an equation with three singularities will suffice. Two wide classes of probability distributions are defined as solutions of such a differential equation, one for continuous distributions and one for discrete distributions. These two classes contain as members all the distributions which are normally considered as of importance in Mathematical Statistics. In the continuous case the probability functions are solutions of the relevant second order equation, while in the discrete case the probability generating functions are solutions there-of. By defining appropriate multidimentional extensions corresponding differential equations are obtained for continuous and discrete multivariate distributions. Dit is bekend dat die opiossings van tweede-orde lineere differensiaalvergelykings met hoogstens vyf singulariteite ’n fundamentele rol speel in die Wiskundige Fisika. In hierdie artikel word aangetoon dat dieselfde bewering ook geld vir die Wiskundige Statistiek maar met die verskil dat ’n vergelyking met drie reguliere singulariteite voldoende is. Twee wye klasse van waarskynlikheidsverdelings, een vir kontinue verdelings en een virdiskrete verdelings, word gedefinieer as opiossings van so ’n differensiaalvergelyking. Daarna word aangetoon dat al die verdelings wat normaalweg in die Wiskundige Statistiek as van belang geag word in hierdie klasse bevat word. In die geval van kontinue verdelings is die waarskynlikheidsfunksies opiossings van die genoemde differensiaalvergelyking terwyl vir diskrete verdelings die waarskynlikheidsvoortbringende funksies wel opiossings van so ’n vergelyking is. Deur geskikte meerdimensionale uitbreidings te definieer volg ooreenkomstige differensiaalvergelykings vir kontinue en diskrete meerveranderlike verdeling. | |
| Publisher | AOSIS | |
| Date | 1985-03-18 | |
| Identifier | 10.4102/satnt.v4i1.1012 | |
| Source | Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie; Vol 4, No 1 (1985); 18-24 Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie; Vol 4, No 1 (1985); 18-24 2222-4173 0254-3486 | |
| Language | eng | |
| Relation |
The following web links (URLs) may trigger a file download or direct you to an alternative webpage to gain access to a publication file format of the published article:
https://journals.satnt.aosis.co.za/index.php/satnt/article/view/1012/2088
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