Abstract
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.
Highlights
It is well-known that the solution o f a second order linear differential equation with at m ost fiv e singularities plays a fu n d a m en ta l role in M athem atical Physics
INLEIDING Reeds aan die einde van die vorige eeu het wiskundiges soos Klein en Bocher aangetoon dat al die lineere differen siaalvergelykings, wat toe van belang was en tans nog in die Wiskundige Fisika van belang is, spesiale gevalle o f konfluente gevalle is van ’n algemene tweede-orde differensiaalvergelyking met vyf singu lariteite ai, a2, aj, a4 en
Dit is ook bekend dat ’n omvattende klas van diskrete verdelings, die sogenaamde hipergeometriese klas, voortgebring word deur K2F|(a;b;c;t), waar K ’n konstante is en 2Fi( ) ’n oplossing is van die hipergeometriese vergelyking, naamlik: t(l - t ) ^ F ( t ) + [ c - ( a + b + l)t];^F{t) - ab F(t) = 0 dt at
Summary
3.1 Diskrete eenveranderiike verdelings 3.1.1 Die eenveranderiike hipergeometriese fam ilie van verdelings. En is gevolglik ’n oplossing van die hipergeometriese vergelyking (6) met a = 1, b = a /d , c = ^ en x = dt. Wanneer d-*0, kan aangetoon word dat die konfluente hipergeometriese vorm. Dit is welbekend dat (15) die wvf is van die hiper-Poissonstelsel van verdelings wat bestaan uit die sub-Poisson-, Poisson- of super-Poissonverdeling al n a /} < l,/3 = l o f/J> l. Hierdie sogenaamde hipergeometriese faktoriaalm om entverdelings is oplossings van die hiper geometriese vergelyking, met veranderlike en para meters soos aangedui. 1= 1 en waar die param eters in (17) sodanig is dat die terme van die reeks eindig en nie-negatief bly en sodat F,( ) konvergeer vir |tj| < 1; i= 1 , 2 , . 224) en dit kan geredelik aangetoon word dat (17) ’n oplossing is van die parsiele differensiaalvergelykings (kyk Steyn'*): (16). Ord se stelsel omvat gevolglik nog meer as die hipergeometriese stelsel m aar bly steeds binne die oplossings van Riemann se vergelyking
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