A note on the sum of uniform random variables

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1 A ote o the sum of uiform radom variables Aiello Buoocore, Erica Pirozzi, Luigia Caputo To cite this versio: Aiello Buoocore, Erica Pirozzi, Luigia Caputo. A ote o the sum of uiform radom variables. Statistics ad Probability Letters, Elsevier, 29, 79 (9), pp.292. <.6/j.spl >. <hal > HAL Id: hal Submitted o 4 Mar 2 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Accepted Mauscript A ote o the sum of uiform radom variables Aiello Buoocore, Erica Pirozzi, Luigia Caputo PII: S67-752(9)244-2 DOI:.6/j.spl Referece: STAPRO 546 To appear i: Statistics ad Probability Letters Received date: 29 Jue 29 Accepted date: 3 Jue 29 Please cite this article as: Buoocore, A., Pirozzi, E., Caputo, L., A ote o the sum of uiform radom variables. Statistics ad Probability Letters (29), doi:.6/j.spl This is a PDF file of a uedited mauscript that has bee accepted for publicatio. As a service to our customers we are providig this early versio of the mauscript. The mauscript will udergo copyeditig, typesettig, ad review of the resultig proof before it is published i its fial form. Please ote that durig the productio process errors may be discovered which could affect the cotet, ad all legal disclaimers that apply to the joural pertai.

3 Mauscript Clic here to view lied Refereces Abstract A Note o the Sum of Uiform Radom Variables Aiello Buoocore,a, Erica Pirozzi a, Luigia Caputo b a Dipartimeto di Matematica e Applicazioi, Uiversità di Napoli Federico II Via Citia, 826 Napoli, Italy b Dipartimeto di Matematica, Uiversità di Torio Via Carlo Alberto, 23 Torio, Italy A iductive procedure is used to obtai distributios ad probability desities for the sum S of idepedet, o equally uiform radom variables. Some ow results are the show to follow immediately as special cases. Uder the assumptio of equally uiform radom variables some ew formulas are obtaied for probabilities ad meas related to S. Fially, some ew recursive formulas ivolvig distributios are derived. Key words: iductio, recursive formulas 2 MSC: 6G5, 6E99. Itroductio The problem of calculatig the distributio of the sum S of uiform radom variables has bee the object of cosiderable attetio eve i recet times. The motivatio ca be ascribed to various reasos such as the ecessity of hadlig data draw from measuremets characterized by differet level of precisio (Bradley ad Gupta, 22), or questios appearig i chage poit aalysis (Sadooghi-Alvadi et al., 29), or, more i geeral, the eed of aggregatig scaled values with differig umbers of sigificat figures (Potuscha ad Müller, 29). It appears that this problem has bee tae up first i Olds (952), where by somewhat obscure procedures formulas for the probability desity fuctio of S ad its distributio fuctio are derived. A accurate bibliography of articles published i the last cetury is foud i Bradley ad Gupta (22), where the authors also obtai the probability desity fuctio of S by o probabilistic argumets, amely via a complicated aalytical iversio of the characteristic fuctio. Such a procedure was successively ad successfully simplified i Potuscha ad Müller (29), where agai o trace of probabilistic argumets is preset. A attempt to achieve the same results by a simpler procedure appears i Sadooghi-Alvadi et al. (29) where a give fuctio is assumed to be the uow probability desity fuctio, the proof of the correctess of such a asatz beig that its Laplace trasform coicides with the momet geeratig fuctio of S. Quite differetly, the preset ote icludes a ovel proof of the above cited results (Propositio 2.). This is based o a iductive procedure, suitably adapted to our geeral istace, used by Feller (966) for the case of idetically distributed variables, that further pipoits the usefuless of iductio procedures i the probability cotext. (See also Hardy et al. (978) for some more illumiatig examples.) I the case of idetically distributed radom variables, some results cocerig certai probabilities ad meas of radom variables related to S are obtaied (Lemma 3., Theorem 3., corollaries 3. ad 3.2, Propositio 3.4), as well as certai recurrece relatios that are remiiscet of those holdig for Stirlig umbers (Propositios 3.5, 3.6, 3.7). Correspodig author; Tel ; Fax addresses: aiello.buoocore@uia.it (Aiello Buoocore), erica.pirozzi@uia.it (Erica Pirozzi), luigia.caputo@uito.it (Luigia Caputo) Preprit submitted to Statistics ad Probability Letters Jue 29, 29

4 The geeral case Let X } N deote a sequece of uiform distributed idepedet radom variables ad deote S = i= X i. Without loss of geerality we assume that X U(, a ) with a positive real umbers. By adoptig a suitably modified procedure due to Feller (966) we shall obtai the probability desity fuctio f (x) ad the distributio fuctio F (x) of S for all N. The startig poit is to write F X (x) = x (x a ) a, N, x R, () where (x c) = maxx c, }, c R. Next we shall mae use of x [ (y c) ] [ dy = (x c) ], N, c R. (2) I additio we ote that, by covolutio, probability desity fuctios ad distributio fuctios are related as follows: f (x) = Claim 2.. Oe has ad a f (x y)f X (y) dy = F (x) F (x a ) a, N, x R. (3) F (x) = x (x a ) a, x R (4) f 2 (x) = x (x a ) (x a 2 ) [x (a a 2 )] a a 2, x R. (5) Proof. It follows from () writte for S X, ad from (3). Claim 2.2. Oe has ad F 2 (x) = (x ) 2 [(x a ) ] 2 [(x a 2 ) ] 2 [x (a a 2 )] } 2 2a a 2, x R (6) f 3 (x) = (x ) 2 [ (x a ) ] 2 [ (x a2 ) ] 2 [ (x a3 ) ] 2 [x (a a 2 )] } 2 [x (a a 3 )] } 2 [x (a 2 a 3 )] } 2 [x (a a 2 a 3 )] } 2 } (2a a 2 a 3 ), x R. Proof. Eq. (6) follows from (5) ad (2). From (6) ad (3) oe the obtais Eq. (7). Claims 2. ad 2.2 lead us to ifer a possible geeral forms of the distributio fuctio of S ad of the probability desity fuctio of S, as specified i the followig Propositio. Propositio 2.. The distributio fuctio F (x) of S ad the probability desity fuctio f (x) of S are give by, respectively: } F (x) = (x ) ( ) ν [x (a j a j2 a jν )] },! A ν= 2 (7) N, x R (8)

5 ad f (x) = (x )! A ν= ( ) ν } [x (a j a j2 a jν )] }, N, x R. (9) Proof. We proceed by iductio. Claims 2. ad 2.2 show that Eqs. (8) ad (9) hold for = ad = 2. Let us ow assume that they hold for = r ad prove that they also hold for = r. To this purpose, we re-write Eq. (9) for = r ad x = y ad the itegrate both sides over (, x). By virtue of (2), Eq. (8) with = r the follows. To obtai Eq. (9) for = r we mae use of (3) ad of the just obtaied expressio of F r (x). Hece, f r (x) = r! A r (x ) r ( ) ν ν= [x (a j a j2 a jν )] } r [ (x a r ) ] r () } ( ) ν [x (a j a j2 a jν a r )] } r. ν= Eq. () idetifies with Eq. (9) writte for = r sice the curly bracets cotais all ad oly all the followig terms:. [(x) ] r ; 2. [(x a ) ] r, [(x a 2 ) ] r,..., [(x a r ) ] r ; 3. for < ν r ( ) ν ( ) ν r ( ) ν r j ν =j ν 2 r [x (a j a j2 a jν )] } r [x (aj a j2 a jν a r ) ] } r [x (a j a j2 a jν )] } r ; ( ) [x r (a a 2 a r ] } r. 43 This complete the iductio A special case Let us assume that the radom variables i X } N are idetically distributed. Propositio 3.. Whe a = a > for all N the ad F (x) =! a ( [(x ( ) ν) ν νa) ], N, x R () ν= ( ) [(x f (x) =! a ( ) ν νa) ], N, x R. (2) ν ν= 3

6 46 47 Proof. Eq. () follows from (8) after otig that ow A = a ad that a j a j2 a jν = νa for ν =,,...,. Ideed, i the sum o ν i (8), the term i curly bracet becomes [(x νa) ], so that [x (a j a j2 a jν )] } ( ) = [(x νa) ]. ν Eq. (2) follows from (9) by a similar argumet. Hereafter, for simplicity we shall tae a = a = for all N. The, from Eqs. (3) there follows f (x) = F (x) F (x ), N, x R (3) so that f () = F () F ( ), N,,,..., }, (4) whereas from Eqs. () ad (2) oe obtais F () =! ad f () = ( )! ( [( ( ) ν) ν ν) ], N,,,..., } (5) ν= ( [( ( ) ν) ν ν) ], N,,,..., }. (6) ν= Propositio 3.2. Whe a = a = for all N the F (x) = f (x j ), N,, 2,..., }, x. (7) j= Proof. Startig from (3), by iteratio it follows that F (x) = f (x) F (x ) = f (x) f (x ) F (x 2) = = f (x j ) F (x ). j= Sice x, oe has F (x ) =, which completes the proof. Propositio 3.3. Whe a = a = for all N the F (x) dx = F (), N,, 2,..., }. (8) Note that Eqs. () ad (2) obtaied by us as a special case of (8) ad (9) are i agreemet with a result due to Feller (966). 4

7 Proof. Maig use of (7) oe obtais F (x) dx = = j= f (x j ) dx = F (x j ) j= [F (j) F (j )] = F () F (). j= The proof is the a cosequece of F () = for all N. Cosider ow the evet S, = S } ad let P, := P (S, ). From (4) it follows that P, = F () F ( ) = f (), N,, 2,..., }. (9) Lemma 3.. Whe a = a = for all N the P (S, S, ) = F () F ( ), N,, 2,..., }. (2) Proof. Let N ad. The, P (S, S, ) = P (X S, S, ) = T f X (x)f (y) dx dy where T deotes the domai i the x-y plae defied by < x < ad < y < x. Hece, by itegratio alog the y-axis from to x, for all x (, ) we obtai P (S, S, ) = Eq. (2) follows from (2) ad (8). = dx x Lemma 3. will be used to prove the followig theorem. Theorem 3.. Whe a = a = for all N the f (y) dy = F ( x) dx F ( ) F (x) dx F ( ). (2) P (S S, ) =, N,, 2,..., }. (22) Proof. Let =. N, from (5) there follows F () = /!, whereas F () =. Hece, maig use of (9) oe obtais P (S S, ) = P (S, S, ) = F () F ()! = P, F () F () ( )! =. (23) This proves (22) for =. From (23) it follows that P (S S, ) = E [P (X y S,, S = y)] = E [S S, ] which ultimately implies E [S S, ] =. Sice X, X 2,..., X are uiform iid radom variables, the mea of each of them coditioal o S, is /( ). Hece, give that S, occurs, the meas of S, S 2,..., S partitio [, ] ito equally wide itervals. Therefore, for <, if S, occurs, the iterval that is partitioed ito equally wide itervals is ow [, ]. This implies that X caot exceed /( ) to isure that S remais below. 5

8 Corollary 3.. Whe a = a = for all N the Proof. Due to Theorem 3. oe has E [S S, ] =, N,, 2,..., }. (24) P (S S, ) = E [P (X y S,, S = y)] = E [S S, ] which ultimately yields Eq. (24). Corollary 3.2. Whe a = a = for all N the Proof. Sice Eq. (25) follows from (24). E [ S S, ] = P,, N,, 2,..., }. (25) E [S S, ] = E [ S S, ] Propositio 3.4. Whe a = a = for all N the = P,, f () = ( ) F (), N,, 2,..., }. (26) = Proof. Let N. From (9) ad (25) we obtai 2 = E [S ] = Maig use of (8), we are easily led to Hece, E [S ] xf (x) dx = xf (x) E [ ] S S, = = F (x) dx = Eq. (26) fially follows by equatig the right had sides of (27) ad (28). The forthcomig recursive formulas are a cosequece of Theorem 3.. f (). (27) = = F (x) dx = F (). = 2 = F (). (28) = Propositio 3.5. Whe a = a = for all N the F () = F () F ( ), N,, 2,..., }. (29) 6

9 Proof. Let N ad. From (9) ad (2) oe obtais or P (S, S, ) = P, P (S, S, ) = F () F ( ) F () F ( ), O the other had, P (S, S, ) = F () F (). (3) P (S, S, ) = P (S S, ) P, = [ P (S S, )] P,. Hece, from (9) ad (22) oe derives P (S, S, ) = F () F ( ) F () F ( ). (3) Eq. (29) the immediately follows after equatig the right had sides of (3) ad (3). Note that (29) trivially holds also for =, yieldig =. Remar 3.. Sice E [ S S, ] = Eq. (25) ca be alteratively obtaied via (29). xf (x) dx = xf (x) Propositio 3.6. Whe a = a = for all N the = F () ( )F ( ) F (). F (x) dx P, = P, P 2,, N,, 2,..., }. (32) Proof. Let N ad. By differece of Eqs. (29) writte for ad for, oe obtais P, = P, F ( ) P, F ( 2), whece (32) follows after otig that F ( ) F ( 2) = P,. Propositio 3.7. Whe a = a = for all N the f () = f () f ( ), N,, 2,..., }. (33) Proof. Let N ad. From (9) ad (32) it follows that f () = P, = P, P,. By maig agai use of (9), Eq. (33) is fially obtaied Acowledgemets We wish to tha Professors R. Johso ad L.M. Ricciardi for helpful commets. 7

10 Refereces Bradley, D.M., Gupta, R.C., 22. O the distributio of the sum of o-idetically distributed uiform radom variables. A. Ist. Statist. Math. 54 (3), Feller, W., 966. A itroductio to probability theory ad its applicatios. Vol. II. Wiley, New Yor. Hardy, G.H., Littlewood, J.E., Pólya, G., 978. Iequalities. Cambridge Uiversity Press, Lodo. Olds, E.G., 952. A ote o the covolutio of uiform distributios. A. Math. Stat. 23, Potuscha, H., Müller, W.G., 29. More o the distributio of the sum of uiform radom variables. Stat. Papers 5, Sadooghi-Alvadi, S.M., Nematollahi, A.R., Habibi, R., 29. O the distributio of the sum of idepedet uiform radom variables. Stat. Papers 5,