# A Recursive Formula for Moments of a Binomial Distribution

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2 The th colum of the variable X refers to the evet of drawig white balls out of a bowl cotaiig a total of equally umbered white ad blac balls I this drawig scheme a particular case of the biomial or Beroulli scheme we require that a extracted ball is reitroduced i the bowl after each drawig Goig bac to the computatio of the first momet 2, it is easy to see that MX = 1 2 =1 1 1 = = 2 3 The trasitio from the first to the secod equality follows from the well-ow Newto biomial idetity 2 q = q = q q r r=0, q N After explaiig the calculatio of 2, as a couter-questio from the studets, it was ased whether the same id of argumet would give a simplified aswer to M 2 X or eve, say, M 2003 X? Ad how about a aswer i closed form for M p X, p 1, i geeral? I what follows, we pla to aswer satisfactorily, we hope the former questio by displayig a recurrece relatio for the geeral p-momets The reader should ote that the recursive formula is useful for calculatios usig pecil ad paper as log as p is i a relatively small rage Observe also that, eve for the particular case of X i discussio, the recursio does ot fall ito a very ice shape Most of the studets would rarely thi of usig pecil ad paper for such computatios cosiderig the sigificat use of software such as STS whe teachig a statistics class However, our decisio of writig this short ote was made by a eed of showig our studets what we as mathematicias do i certai situatios: we try to prove rigorously ad i most geerality the observatios we mae by looig at several particular cases displayed o a computer scree Our mai goal is to show the followig result: Propositio Let p 1 The a M p X = M p 1 X 1 2 M p 1X 1 ; 4 b M p X = 2 Q p 1, where Q p p 1 is the moic polyomial of degree p 1 i the variable which solves the recursio Q p = 2Q p 1 1Q p 1 1 Proof Let us re-deote S, p = 2 M p X = p =0 Sice p C = p 1 = p 1 1 1, VOL 36, NO 1, JANUARY 2005 THE COLLEGE MATHEMATICS JOURNAL 69

3 we ca write or S, p = =1 p = p 1 =0 = =0 p 1 =1 p S, p = S, p 1 S 1, p 1 ; 5 i the secod sum above we used the covetio 1 0 It is clear that 5 traslates to the p-momets exactly as the recursio a We ow prove part b Newto s biomial formula gives S, 0 = 2 Fromhere ad 5, we get S, 1 = S, 0 S 1, 0 = = 2 1 Note that, i particular, we recover the computatio of the first momet 3 Similarly, oe gets S, 2 = , S, 3 = , etc The formulas obtaied for the first four sums S, p, 0 p 3, suggest that S, p = 2 p Q p 1, 6 where Q p 1 is some polyomial of degree p 1ithevariable with uitary leadig coefficiet Ideed, oe ca prove this usig 5 ad a iductio argumet o p Assumig that 6 holds true, we have S, p + 1 = 2 p Q p p Q p 1 1 where we deoted = 2 p 1 2Q p 1 1Q p 1 1 = 2 p+1 Q p, Q p = 2Q p 1 1Q p 1 1 By our iductive assumptio, Q p 1 = p 1 + lower order terms To complete the iductio step, we oly eed to show that Q p is a polyomial of degree p i the variable with uitary leadig coefficiet To see this, ote that the leadig term of Q p comes from the differece 2 p p 1 = p + p 1 p = p + lower order terms, which proves our claim Observe that 6 ca be rewritte as which is equivalet to the statemet b 2 M p X = 2 p Q p 1, We remar that fidig explicitly the polyomials Q p for a geeral positive iteger p turs out to be a otrivial tas The iterested reader should chec for example that already for M 4 X, the polyomial Q 3 does ot factor completely over 70 c THE MATHEMATICAL ASSOCIATION OF AMERICA

5 2 A T Craig ad R V Hogg, Itroductio to Mathematical Statistics, 5th ed, Macmilla, J L Devore, Probability ad Statistics: for Egieerig ad the Scieces, 4th ed, Broos/Cole, R J Larso ad M L Marx, A Itroductio to Mathematical Statistics ad Its Applicatios, 3rd ed, Pretice Hall, 2001 Proof Without Words From Richard Hammac of Robert Madiso Uiversity ad David Lyos of Lebao Valley College: Theorem A alteratig series a 1 a 2 + a 3 a 4 + a 5 a 6 + a 7 a 8 + coverges to a sum S if a 1 a 2 a 3 a 4 0 ad a 0 Moreover, if s = a 1 a 2 + a 3 ±a is the th partial sum the s 2 < S < s 2+1 Proof S a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 2 1 a 2 a 2+1 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 2 1 a 2 s 2 s 2+1 A differet versio of this result with words is to appear i the Oxford Joural of Teachig Mathematics ad Its Applicatio 72 c THE MATHEMATICAL ASSOCIATION OF AMERICA

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