Application of Generating Functions to the Theory of Success Runs

Transcription

1 Aled Mathematcal Sceces, Vol. 10, 2016, o. 50, HIKARI Ltd, htt://dx.do.org/ /ams Alcato of Geeratg Fuctos to the Theory of Success Rus B.M. Bekker, O.A. Ivaov ad V.V. Ivaova St. Petersburg State Uversty, Uverstetskaya ab., 7-9, St. Petersburg, Russa Coyrght c 2016 B.M. Bekker, O.A. Ivaov, ad V.V. Ivaova. Ths artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese, whch ermts urestrcted use, dstrbuto, ad reroducto ay medum, rovded the orgal work s roerly cted. Abstract I the reset aer, we geeralze the results of the aers [2] ad [4] ad gve a ew roof of the exlct formula for the geeratg fucto of the umbers of trals before gettg k cosecutve successes. Mathematcs Subject Classfcato: 12J10 Keywords: Geeratg fucto, Beroull scheme 1 Itroducto We cosder the followg well-kow roblem: what s the exected umber of co tosses before two cosecutve heads are obtaed? A smlar questo ca be asked about a far sx-sded de: how may tmes do you have to roll before gettg a gve umber k of cosecutve sxes? Formally, the roblem ca be stated as follows. We cosder the Beroull scheme of trals whch success S occurs wth robablty ad falure F occurs wth robablty q = 1. Let X be the radom varable whose values are the umbers of trals before gettg a success ru of legth k. The geeratg fucto G X of X s well kow (see e.g., [3, Ch. XIII, Sec.7]. Noetheless, ths roblem stll attracts atteto of mathematcas. I the aers [2] ad [4], t s show that, the case = 1/2, the geeratg fucto G X s coected wth the geeratg fucto of the so-called k-fboacc umbers. Sce the geeratg fucto for these umbers s costructed a stadard way, oe has a ew roof of

2 2492 B.M. Bekker, O.A. Ivaov, ad V.V. Ivaova the formula for the geeratg fucto questo. I the reset aer, we geeralze these results to the case of a arbtrary. 2 Geeratg fucto of the umber of trals We cosder the sequeces of trals wth o success rus of legth k. Each such sequece ca be rereseted as a sequece of F s ad S s wth o k successve S s. For each = 0, 1,..., k 1, we deote by H the evet that a sequece of trals the outcome S occurs at the last trals. I artcular, H 0 meas that the outcome F occurs at the last tral. Let u ( be the robablty of H. For = 0, 1,..., k 2, the trasto {}}{}... F {{ S... S} +1 {}}{.}.. F S {{... SS} +1 haes wth robablty, hece u (+1 +1 = u ( q = 1, we have +1 = (1 ( + u (1 the followg system of recurrece relatos: +1 = (1 ( + u (1 u (1 +1 =, u (2 +1 = u (1,... u (k 1 +1 = u (k 2. For k 1, we ut v = u (k 1. The u (1. Sce the robablty of F s u (k 1. Thus, we obta + + u (k 1 = 1 u(1 +1 = 1 u ( = = 1 u (k 1 k 1 +k 1 = 1 = 1 u(2 +1 = = 1 u (k 1 2 +k 2 = 1 v k 2 +k 2,... u (k 1 = v, whch mles that +1 = 1 k 1 v +k, v +k = (1 k 1 ( v+k 1 k 1 + v +k 2 k 2, k 1 v +k 1, + + v. Dvdg both sdes of the equato obtaed by, we obta ( v +k v+k 1 = (1 +k 1 + v +k 2 +k v. +k 2

3 Alcato of geeratg fuctos 2493 Puttg x = v, we ca rereset the last relato the form hece x +k = (1 (x +k 1 + x +k x, x +k = 1 (x +k 1 + x +k x. (1 If = 1/2, the relato (1 has the form x +k = x +k 1 + x +k x, whch, for k = 2, cocdes wth the recurrece relato for Fboacc umbers. Relato (1 s roved for k 1. For = 0, 1,..., k 2, we ut x = 0 ad calculate the values of x at = k 1, k,..., 2k 1. By defto, we = P ( H k 1. The evet H k 1 k 1 cossts of Therefore, ts robablty s equal to k 1, ad so, x k 1 = 1. If k 2k 1, the the evet H k 1 cossts of have x = v = u (k 1 oe sequece S... S of legth k 1. k {}}{ the sequeces of the form }... F{{ S... S}, where dots at the begg mea F s or S s. Cosequetly, P ( H k 1 = (1 k 1, ad x = (1 k 1 for = k, k + 1,..., 2k 1. Lemma 2.1. Relato (1 s vald for all tegers 0. Proof. For = 0, we have x k = (1 / x k 1. Ths relato s vald sce x k = (1 / ad x k 1 = 1. If 1 k 1, the k k 2k 1. Therefore, x +k = (1 1. Let us calculate the rght-had sde of equato (1. We have x +k 1 + x +k x = = 1 = = 1. Cosequetly, the rght-had sde of equato (1 s equal to (1 1 = (1 1 = x +k. Lemma 2.2. The geeratg fucto of the sequece x s gve by the formula ϕ(t = t k (t + t t k = (tk 1 tk (1 t k+1 t +. (2

4 2494 B.M. Bekker, O.A. Ivaov, ad V.V. Ivaova Proof. By defto, ϕ(t = =k 1 x t. Removg brackets (x k 1 t k 1 + x k t k + ( 1 1 (t + t2 + + t k, we obta by relato (1 that x k 1 t k 1 + ( x k 1 whch mles equato (2. x k 1 t k + ( x k+1 1 (x k + x k 1 t k ( x 2k 2 1 (x 2k x k 1 t 2k 2 + ( x +k 1 (x +k x t +k + = t k 1, Lemma 2.3. We have G X (t = tϕ(t, where ϕ(t s the geeratg fucto of the sequece x. Proof. We ote that the robablty that k successve evets h occur the frst tme mmedately after the th tral Beroull s scheme s the roduct of ad the robablty of H 1. k 1 Thus, P (X = = P ( H 1 k 1 = u k 1 1 = v 1 = x 1. Therefore, G X (t = P (X = t = =k x 1 t = t =k =k 1 x (t = tϕ(t. I cocluso, we derve exlct formulas for the terms of the sequece x. We cosder the olyomal g(z = z k 1 (z k 1 + z k Let z 1, z 2,..., z k be the comlex roots of g(z, ad let a = k =1 z g (z. Sce g(z = (z z 1 (z z 2... (z z k, we have the relatos a 0 = a 1 = = a k 2 = 0 ad a k 1 = 1. Sce z are the roots of g(z, we obta z +k = z z k = 1 ( z +k 1 + z +k z,

5 Alcato of geeratg fuctos 2495 whch mles the relato a +k = 1 (a +k 1 + a +k a. Therefore, the umbers a satsfy the same kth order recurrece relatos as the umbers x do. Takg to accout that the frst k terms of the sequeces questo cocde, we see that x = a for all 0. As a corollary, we obta the followg exlct formula for the terms of the sequece x : x = k =1 z g (z. (3 Now we use formula (5 from the aer [1]. Let w be the comlete homogeeous olyomal of degree k varables. We ut, by defto, w 0 = 1 ad w 1 = w 2 = = w 1 k = 0. By Theorem 1 of the aer [1], we have the relato k z +k 1 = w g (z 1, z 2,..., z k, (z =1 from whch we obta by (3 that x = w k+1 (z 1, z 2,..., z k. Ackowledgemets. The frst amed author was artally suorted by the Russa Foudato for Basc Research (grat o Refereces [1] B. M. Bekker, O. A. Ivaov ad A. S. Merkurjev, A Algebrac Idetty ad the Jacob Trud Formula, Vestk St. Petersburg Uversty: Mathematcs, 49 (2016, o. 1, 1-4. htt://dx.do.org/ /s [2] L.M. Chaves ad D.J. de Souza, Watg tme for a ru of N successes Beroull sequeces, Rev. Bras. Bom., São Paulo, 25 (2007, o. 4, [3] W. Feller, A Itroducto to Probablty Theory ad Its Alcatos, Vol. 1, 3rd ed., Joh Wley & Sos, New York, [4] A.N. Phlou ad A.A. Muwaf, Watg for Kth cosecutve success ad the Fboacc sequece of order K, Fboacc Quart., 20 (1982, o. 1, Receved: July 6, 2016; Publshed: August 14, 2016