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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 is to derive the

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

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

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