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Classroom Note an Inductive Derivation of Stirling Numbers Of View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Crossref c Journal of Applied Mathematics Decision Sciences Reprints Available directly from the Editor Printed in New Zealand Classro om Note An Inductive Derivation of Stirling Numbers of the Second Kind and their Applications in Statistics y ANWAR H JOARDER Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran Saudi Arabia MUNIR MAHMOOD Department of Econometrics Monash University Clayton Victoria Australia Abstract An inductive metho d has b een presented for nding Stirling numb ers of the second kind Applications to some discrete probability distributions for nding higher order moments have b een discussed Keywords Stirling numb ers of the second kind raw moments factorial moments Intro duction n Stirling numbers of the second kind are used to express x where n is a non negative integer as sums of pow ers of factorial p olynomials not higher than the n nth For every nonnegativeinteger n the function x can b e expressed as n x S n x S n x S n nx n where x xx x r is the factorial p olynomial of order r r r n and S n Sn Sn n are called Stirling numb ers of the sec n r isthenumber of ways to distribute n ond kind It may b e mentioned that S distinguishable balls into r indistinguishable urns with no urn empty These numb ers are called Stirling numb ers of the second kind after the British mathematician James Stirling In this note we present an inductive pro of to deriveStirlingnumb ers of the second kind For a combinatorial pro of the reader is referred to Rob erts pp We also demonstrate the application of Stirling numb ers in calculating moments of some discrete distributions The usual metho d of deriving raw mo ments of higher order of an integervalued random variable is to derive the moment y The work was done in part while the rst author was at the University of Sydney Australia A H JOARDER AND M MAHMOOD generating function and then dierentiate as many times as the order of the moment required In this pap er wetakeanadvantage of factorial moments which are easily derived for integervalued random variables These moments can b e combined with the help of Stirling numb ers of the second kind for deriving raw moments and hence central moments of integervalued random variables The advantage of the metho d presented here for calculating raw moments or central moments of a distribution is that it avoids using derivatives of higher order The Description of the Metho d Tables showing Stirling numb ers up to n is wellknown see eg Beyer p These tables can be prepared by simple algebraic manipulations Supp ose that wewant to calculate Stirling numb ers in for n so that x S x S x S x S x S xx S xx x Equating the co ecients of x x and x we have the following triangular system of equations S S S S S S If the Solving the equations wehave S S and S ab ove metho d of calculating Stirling numbers is rep eated for n etc then it is easy to prepare a table showing Stirling numbers for dierent n and r The following wellknown recurrence relation see eg Beyer p is similar to the relation among binomial co ecients in Pascal triangle except a multiple to the second term S n r S n r rSn rr n with S n S n n and S n r rnFor a simple algebraic pro of see Berman and Fryer pp In this section we describ e that the wellknown form ula for a Stirling number of the second kind given by r X r i n r i S n r i r i r X r n r i i i r i AN INDUCTIVE DERIVATION OF STIRLING NUMBERS OF THE SECOND KIND Rob erts pp where r n and n is any nonnegativeinteger can b e derived from by induction The last line in the identity follows bychanging the index of summation i to j where j r i By the rep eated use of it is easy to check that the Stirling numb ers of the second kind S n r r can b e written as n X i n S n S n i n i n X i i n S n S n i n X i ni n n i n X i S n S n i n i n X i ni ni f g i n n n X i n i i i r where is the combination of r things taking i at a time i We assume that the identity is true and try to provethe identity in The identity holds for S n and S n It maybeproved by induction that for all real numb ers r n X n i n r i n i i Ruiz and consequently S n n Applying induction hyp othesis for n andsor n wehave r X r n r i S n r i i r i r X r r i n ir i i r i A H JOARDER AND M MAHMOOD r X r n r i i i r i r X i r r i n i i r r i and r X r r i n rSn r r i i r i r n X r r n r i i i r r i so that S n r S n r r S n r r n X r i r r i n i i r r r i The summand in the second term in the ab ove expression is exactly the same as the rst term if i r and consequently the two terms add to the expression in The pro of is thus complete The ab ove pro of has p edagogic values esp ecially to readers who want to avoid combinatorial pro of Applications Stirling numbers connect the factorial moments to raw moments and vice versa Taking exp ectations in the b oth sides of with n and r replaced by r and i resp ectively we have the general relationship between rawmoments and factorial moments as r X S ri r i i r where E X is the r th r n rawmoment and E X r r r is the r th r n factorial moment Harris p rst used this formula clearly in the context of statistics though it has not b een p opular Stuart and Ord p presented the formula for r and also mentioned that Frischgave general formulae showing factorial moments ab out one p oint in terms of ordinary moments ab out another and vice versa However AN INDUCTIVE DERIVATION OF STIRLING NUMBERS OF THE SECOND KIND formulae given by Frisch involve Bernoulli p olynomials and are not easy to use The central moments are connected to the raw moments by the following relation r X r i i r r i i i The factorial moments of integer valued distributions eg binomial and negative binomial distributions can b e easily derived Then these moments can b e combined by and to deriveraw and central moments without having to resort to any dierentiation Some examples are presented b elow to illustrate the application Example Let us calculate the fourth raw moment of the binomial distribution B n pby the use of Here wehave S S S S It follows from that S S S and S The ith factorial moment of the binomial distribution is easily shown i to b e n p i n Therefore the fourth raw moment of the binomial i i distribution is given by np n p n p n p Example Consider deriving the moments of the geometric distribution Let X be the numb er of trials prior to the rst success with probability density function pdf x P X xq p p q p x i It can b e easily veried that iqp i By the use of the i formula wehave q p qpqp S S qpqp qp qpqp qp qp Nowby the use of the moment relationship in followed by some algebraic manipulations we nd that the second third and fourth order central moments of the geometric distribution are qp q q p q q q p A H JOARDER AND M MAHMOOD Example Let us derive moments of the negative binomial distribution Let X b e the numb er of trials prior to sth s success with pdf s x x s P X x q p q p px x i It can b e easily veried that s i qp i i i By the use of the formula wehave sq p sq sq p sqps qp s qp sqps qp s qp s qp Then by the use of we nd that the second third and fourth order central moments of the negative binomial distribution are sq p sqpsqp sqp s q q p sqpssqp s sqp s sqp sq s q q p cf Evans M et al p Acknowledgments The authors thank Dr David Easdown the University of Sydney and Dr M Iqbal King Fahd UniversityofPetroleum and Minerals for their constructive suggestions on an earlier draft of this pap er The authors are also grateful to the editor Dr Mahyar Amouzegar and an anonymous referee for their constructive suggestions that considerably improved the presentation of this pap er References Berman G and Fryer KD Introduction to Combinatorics Academic Press New York Beyer WH Handbook of Mathematical Sciences CRC Press Florida Evans M Hastings N and Peaco ckB Statistical Distributions John WileyNewYork Harris B Theory of Probability AddisonWesley Pub Co Massachusetts AN INDUCTIVE DERIVATION OF STIRLING NUMBERS OF THE SECOND KIND Rob erts FS Applied Combinatorics Prentice Hall New Jersey Ruiz SM An algebraic identity leading to Wilsons theorem Mathematical Gazette Stuart A and Ord JK Kendals Advanced Theory of StatisticsVol Charles Grin and Co Ltd London Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 The Scientific International Journal of World Journal Differential Equations Hindawi Publishing Corporation Hindawi Publishing Corporation 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