A probabilistic proof of a binomial identity


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1 A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two differet ways, yieldig two differet expressios for the same quatity. The goal of this ote is to give a simple (ad iterestig) probabilistic proof of the biomial idetity ( ) ( 1) θ, for all θ > 0 ad all N. (1) 0 If oe is oly cocered with givig a proof of this equality, other proofs tha the probabilistic oe give below may be more atural. For istace, a proof may be give by idetifyig the left side of (1) as the evaluatio of a hypergeometric fuctio 2 F 1 (, θ; θ + 1 1) ad the applyig the ChuVadermode formula [2, equatio (1.2.9)] to obtai the right side of (1). Aother approach would be to use the Rice itegral formulas [1, 3] to equate the left side of (1) with a complex cotour itegral that ca be see to equal the right side of (1). These approaches give short proofs of (1), but they both use a good deal of advaced mathematics. With a bit of wor, oe ca also obtai a elemetary proof of (1) usig oly basic properties of the biomial coefficiets ad mathematical iductio. The proof of (1) give below arose ot i a search for a ew proof of this idetity, but as a result of some idepedet probability research... ad a cluttered des. Beig uable to fid a probability calculatio I had doe the day before, I sought to repeat the calculatio but obtaied a differet expressio for the same quatity. After some iitial cofusio, I realized that my computatios gave a simple proof of the idetity (1). 1 Probability theory bacgroud. Before givig the probabilistic proof of (1), I will recall some basic facts from probability theory. All of the probability eeded for this paper ca be foud i a basic udergraduate probability boo such as [4]. Recall that a radom variable Y has a expoetial distributio with parameter λ if { 1 e λy y 0 P (Y y) 0 y < 0. 1
2 The otatio Y Exp(λ) will be used to deote that Y has a expoetial distributio with parameter λ. The followig elemetary fact about expoetial radom variables will be used multiple times i the proof of (1). E[e θy ] λ λ + θ for all θ > λ if Y Exp(λ). (2) Aother basic tool from probability theory that will be eeded is the method of computig probabilities by coditioig. Suppose that A is a evet that depeds o some radom variable Z ad some additioal radomess. It is sometimes easier to compute the probability of the evet A if Z is fixed (that is, by coditioig o Z). The, the probability of the evet A is obtaied by averagig the coditioal probabilities over all values of Z: P (A) E[P (A Z)]. As a example of this, suppose Y Exp(λ) ad Z Exp(µ) are idepedet. The, we ca compute P (Y < Z) by coditioig o Y. Sice P (Y < Z Y y) e µy we see that P (Y < Z) E[P (Y < Z Y )] E[e µy ] where the last equality follows from (2). 2 Probabilistic proof. λ λ + µ, Suppose that X 1, X 2,..., X are idepedet Exp(1) radom variables, ad let X max i X i. Also, let T Exp(θ) be idepedet of the X i (ad thus also idepedet of X). The proof of (1) give below shows that both sides of (1) are equal to the probability P (X < T ). The two differet represetatios of this probability ca both be obtaied by coditioig the first by coditioig o X ad the secod by coditioig o T. 2.1 Coditioig o X. Sice T Exp(θ), the coditioal probability with respect to X is P (X < T X x) e θx. Therefore, P (X < T ) E[P (X < T X)] E[e θx ]. (3) The above expoetial momet of X ca be computed usig the followig alterate represetatio of X. Lemma 1. Suppose that X 1, X 2,... X are idepedet Exp(1) radom variables, ad that X max i X i. The, X has the same distributio as Y, where the Y are idepedet radom variables with with Y Exp(). Remar 1. The above represetatio of X as the sum of idepedet expoetial radom variables is ot stadard material i a udergraduate probability course. However, the proof of Lemma 1 below oly uses two basic facts about expoetial radom variables that typically are part of a udergraduate probability course. 2
3 Proof. The proof of this lemma relys o two basic facts about expoetial distributios. The first is that expoetial distributios are memoryless: if Z Exp(λ) the P (Z > t+s Z > s) P (Z > t). That is, thiig of a expoetial radom variable as the radom amout of time before a evet happes, if the evet has ot occured by time t, the the remaiig amout of time before the evet occurs is still a expoetial distributio with the same parameter. The secod basic fact eeded for the proof of Lemma 1 is that if Z 1, Z 2,..., Z are idepedet Exp(1) radom variables, the mi i Z i Exp(). To see this, ote that idepedece implies ( ) ( ) P mi Z i > t P {Z i > t} P (Z i > t) e t. i i1 Usig these two facts, the proof of Lemma 1 is most easily explaied i the followig way. Suppose that there are lightbulbs i a room that are all tured o at the same time ad that X i is the amout of time util the ith lightbulb fails. The X max i X i is the amout of time util all of the lightbulbs have failed. By the secod fact above, the amout of time util oe of the lightbulbs burs out is a expoetial radom variable with parameter. At this time there are still 1 lightbulbs worig, ad by the first fact above the remaiig lifetime of each of these lightbulbs is a Exp(1) radom variable. Thus X has the same distributio as the sum of a Exp() radom variable ad a idepedet radom variable X that is the maximum of 1 idepedet Exp(1) radom variables. The coclusio of the lemma the follows by iductio o. Applyig Lemma 1 to (3) implies that P (X < T ) E [e θ ] Y i1 E [ e θy ] + θ, (4) where the secod equality follows from the idepedece of the Y ad the last equality follows from (2). 2.2 Coditioig o T. A differet expressio for P (X < T ) ca be obtaied by coditioig o T istead. Sice {X < t} i1 {X i < t} ad the X i are idepedet, it follows that P (X < T T t) P (X i < t) ( 1 e t) ( ) ( 1) e t. i1 3
4 The, taig expectatios with respect to T gives [ ( ] ( ) P (X < T ) E )( 1) e T ( 1) E[e T ] ( ) ( 1) θ, (5) where the secod equality is from the liearity of expected values ad the last equality is from (2). The proof of the biomial idetity (1) is the completed by combiig (4) ad (5). 3 Geeralizatios. Sice this probabilistic proof of (1) was costructed quite by accidet, it is difficult to use this method to prove a give biomial idetity. However, the above method ca be used to discover other iterestig biomial idetities by maig chages to the origial probability beig computed. For istace, let X ad T be as above ad let T 2 T + T where T is a idepedet Exp(θ) radom variable. The, computig P (X < T 2 ) two differet ways will give the followig idetity 0 ( ) ( ) ( 2 ) ( θ ( 1) 1 + ) θ. The left side of the above idetity is obtaied by first coditioig o T 2, while the right side is obtaied by coditioig o X. Eve more geerally, for ay oegative iteger m oe ca obtai the idetity ( ) ( ) m θ ( 1) 0 ( ) m j j θ j ( 1 )( 2 ) ( j ) by computig P (X < T m ) whe T m is the sum of m idepedet Exp(θ) radom variables (ote that T m is a Gamma(m, θ) radom variable). Agai, these idetities could be proved usig the Rice itegral formulas, but the iterested reader may ejoy usig the probabilistic method above to prove these idetities istead. Acowledgmets. The probability calculatio that led to the above proof was doe while doig research supported by Natioal Sciece Foudatio grat DMS I am very grateful to the NSF for this support. 4
5 Refereces [1] P. Flajolet ad R. Sedgewic, Melli trasforms ad asymptotics: fiite differeces ad Rice s itegrals, Theoret. Comput. Sci. 144 (1995) [2] G. Gasper ad M. Rahma, Basic Hypergeometric Series, Ecyclopedia of Mathematics ad its Applicatios, Vol. 35, Cambridge Uiversity Press, Cambridge, [3] P. Kirschehofer, A ote o alteratig sums. Electro. J. Combi., 3 o. 2 (1996) Research Paper 7, idex.php/eljc/article/view/v3i2r7. [4] S. Ross, A First Course i Probability, eighth editio. Pearso Pretice Hall, Upper Saddle River, NJ, Departmet of Mathematics, Purdue Uiversity, 150 N. Uiversity Street, West Lafayette, IN
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