ON POINTWISE BINOMIAL APPROXIMATION

Size: px
Start display at page:

Download "ON POINTWISE BINOMIAL APPROXIMATION"

Transcription

1 Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No , ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece Burapha Uiversity Choburi, 20131, THAILAND Abstract: This paper, we use Stei s method ad w-fuctios to give a result i the biomial approximatio to the distributio of a o-egative itegervalued radom variable, i terms of the poit metric betwee two such distributios together with its o uiform boud. Furthermore, for applicatios, we use the obtaied result to approximate some distributios such as hypergeometric, egative hypergeometric ad Pólya distributios. AMS Subject Classificatio: 62E17, 60F05 Key Words: biomial approximatio, o-uiform boud, poit metric, Stei s method, w-fuctios 1. Itroductio May studies of biomial approximatio via Stei s method have yielded useful results i applicatios of probability ad statistics. The first study of biomial approximatio by Stei s method, for approximatig the umber of oes i the biary expasio of a radom iteger ad for problem of coutig Lati rectagles, was preseted by Stei [6]. Ehm [3] gave lower ad upper bouds of the error i the biomial approximatio of a sum of idepedet idicator radom variables, ad he applied the result to samplig with ad without replacemet. Barbour et al. [1] showed that Stei s method could be applied as well i the biomial cotext as i the Poisso. Soo [5] cosidered this approximatio i coectio with a sum of depedet idicator radom variables, Received: May 4, 2011 c 2011 Academic Publicatios, Ltd. Correspodece author

2 58 K. Teerapabolar, P. Wogkasem ad he applied the result to hypergeometric distributio, radom graphs problem ad the classical occupacy problem. Wogkasem et al. [8] used Stei s method ad w-fuctios to give a error boud o biomial approximatio to a geeralized biomial distributio, ad Teerapabolar [7] used the same tools as i Wogkasem et al. [8] to give a error boud o biomial approximatio to the beta biomial distributio i the recet paper. However, all bouds as metioed above are the total variatio distace bouds. I this paper, we use Steis method ad w-fuctios to give a o uiform boud i the biomial approximatio of a o-egative iteger-valued radom variable for the poit metric betwee the two distributios. Let X be a o-egative iteger-valued radom variable with probability fuctio p(x) = P(X = x) > 0 for every x i the support of X, S(x), ad have mea µ ad fiite variace σ 2 (0 < σ 2 < ). It is well-kow that the distributios of some types of X s ca be approximated by a biomial distributio with parameters ad p provided their parameters are satisfied uder certai coditios. For example, a hypergeometric distributio ca be approximated by a biomial distributio provided that the certai coditios o their parameters are satisfied. Let b( ;,p) = ( ) p q deote a biomial probability fuctio with parameters N ad p = 1 q (0,1) at {0,...,}. If we expect p( ) to be closer to b( ;,p), the it is reasoable to estimate p( ) by b( ;,p). For approximatig p( ) by b( ;,p), a boud for the poit metric betwee p( ) ad b( ;,p) is a criterio for measurig the accuracy of the approximatio. I this study, we derive a o-uiform boud of the error o the metric betwee p( ) ad b( ;,p). The tools for givig our result cosist of the socalled w-fuctios ad Steis method for the biomial distributio, which are i Sectio 2. I Sectio 3, we use Stei s method ad w-fuctios to give the result i terms of the poit metric betwee p( ) ad b( ;,p), ad we give some applicatios of the result of this approximatio by usig the result to approximate some distributios such as hypergeometric, egative hypergeometric ad Pólya distributios, which are i the last sectio. 2. Method I order to give the result for this approximatio, we use the same methodology as i Teerapabolar [7], which cosists of Stei s method ad w-fuctios. For w-fuctios, Cacoullos ad Papathaasiou [2] defied a fuctio w associated

3 ON POINTWISE BINOMIAL APPROXIMATION with o-egative iteger-valued radom variable X i the relatio w(x)p(x) = 1 σ 2 x (µ i)p(i), x S(x) (2.1) i=0 ad, afterwards, Majserowska [4] expressed the relatio (2.1) as the form w(0) = µ σ 2, w(x) = 1 { } σ 2 µ + σ2 w(x 1)p(x 1) x, x S(x) \ {0} (2.2) p(x) ad w(x) 0, x S(x), (2.3) where p(x) > 0 for every x S(x). The followig relatio is a importat property for provig the result, which was stated by Cacoullos ad Papathaasiou [2]. If a o-egative iteger-valued radom variable X is defied as i Sectio 1, the E[(X µ)g(x)] = σ 2 E[w(X) g(x)], (2.4) for ay fuctio g : N {0} R for which E w(x) g(x) <, where g(x) = g(x + 1) g(x). For g(x) = x, we have that E[w(X)] = 1. For Stei s method, we start it by usig Stei s equatio i Barbour et al. [1]. Stei s equatio for the biomial distributio with parameters N ad p (0,1) is, for give h, of the form ( x)pg(x + 1) qxg(x) = h(x) B,p (h), (2.5) where B,p (h) = k=0 h(k)( k) p k q k ad g ad h are bouded real-valued fuctios defied o {0,1,...,}. For A {0,1,...,}, let h A : {0,1,...,} R be defied by h A (x) = { 1 if x A, 0 if x / A. (2.6) By followig Barbour et al. [1] o pp. 189, let g A : N {0} R satisfy (2.5), where g A (0) = g A (1) ad g A (x) = g A () for x.

4 60 K. Teerapabolar, P. Wogkasem For A = { }, {0,...,}, the solutio g x0 = g {x0 } of (2.5) ca be writte as ( x )p x B 0,p(1 h Cx 1 ) if x x( x)q (x 1) 0 < x, g x0 (x) = (2.7) ( x )p x B 0,p(h Cx 1 ) if x x( x)q (x 1) 0 x 1, where C x = {0,...,x}. To prove the result, the followig lemma is also eed. Lemma 2.1. For {0,1,...,} ad x N, let g x0 (x) = g x0 (x + 1) g x0 (x), the we have the followig. ad g x0 (x) g x0 (x) { 1 q p { } mi 1 p q, 1 p+1 q +1 (+1)pq if > 0 { 1 q p { } mi 1 p q, 1 p+1 q +1 (+1)pq if > 0 (2.8) (2.9) Proof. For = 0, it follows from [1] that g 0 is positive ad decreasig i x {1,...,} ad g 0 (x) = 0 for x = 0 ad x. Therefore, we have g 0 (1) g 0 (x) g 0 (x) for every x {1,...,} ad, by (2.7), g 0 (1) = 1 q which implies g 0 (x) 1 q p ad g 0 (x) 1 q p. For > 0, it follows from [1] that g x0 is positive ad decreasig i x { + 1,...,} ad is egative ad decreasig i x {1,..., } ad g x0 (x) = 0 for x = 0 ad x. Therefore, we have that g x0 (x) g x0 ( ) ad g x0 (x) g x0 ( ) ad 1 g x0 ( ) = ( )p = 1 q 1 q k= +1 1 k= k= +1 ( ) p k q k + 1 k q x 0 1 k=0 ( ) p k q k k ( ) p k 1 q +1 k ( + 1 k) + k 1 k( ) ) x 0 1( ) p k q k + p k q k k ( k k=0 x 0 1 k=0 ( ) k p, p k q k

5 ON POINTWISE BINOMIAL APPROXIMATION ad g x0 ( ) = gives ad = 1 p q = k= +1 k= +1 1! k!( k)!( ) pk q +1 k pq + (+1)!(+1 k) k!(+1 k)!( ) pk q +1 k ( + 1 k= +1 = 1 p+1 q +1, k x0 1 k=0 + x0 1 k=0 ) p k q +1 k + 1 q! k!( k)! pk+1 q k pq (+1)!(k+1) (k+1)!( k)! p k+1 q k k=1 g x0 (x) mi{ 1 p q, 1 p+1 q +1 } g x0 (x) mi{ 1 p q, 1 p+1 q +1 }. So, from both cases, (2.8) ad (2.9) are obtaied. ( ) + 1 p k q +1 k k 3. Result The followig theorem shows a result i the biomial approximatio to the distributio of a o-egative iteger-valued radom variable X, i terms of the poit metric ad its o-uiform boud, which is obtaied by Stei s method ad w-fuctios. Theorem 3.1. Let a o-egative iteger-valued radom variable X with p(x) > 0 for every x S(x) ad together with correspodig w-fuctio w(x) be defied as above. The the followig iequalities hold: 1. For = 0, p(0) b(0;,p) 1 q p { E ( X)p σ 2 w(x) + p µ } (3.1)

6 62 K. Teerapabolar, P. Wogkasem ad, if p = µ, the p(0) b(0;,p) 1 q p E ( X)p σ 2 w(x). (3.2) 2. For {1,...,}, { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } { E ( X)p σ 2 w(x) } + p µ (3.3) ad, if p = µ, the { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } E ( X)p σ 2 w(x). (3.4) Proof. Substitutig h by h {x0 }, x by X ad takig expectatio i (2.5), we obtai p( ) b( ;,p) = E[( X)pg(X + 1) qxg(x)], (3.5) where g = g x0 is defied i (2.7) ad E[( X)pg(X + 1) qxg(x)] = E[pg(X + 1) px g(x) Xg(X)] = E[pg(X + 1)] pe[x g(x)] E[Xg(X)] = pe[g(x + 1)] pe[x g(x)] E[(X µ)g(x)] µe[g(x)] = pe[ g(x)] pe[x g(x)] E[(X µ)g(x)] + (p µ)e[g(x)] Sice E[w(X)] = 1 ad g(x) is bouded, the E w(x) g(x) <. Thus, by (2.4), it follows that E[( X)pg(X + 1) qxg(x)] = pe[ g(x)] pe[x g(x)] which, by (3.5), yields σ 2 E[w(X) g(x)] + (p µ)e[g(x)] = E{[( X)p σ 2 w(x)] g(x)} + (p µ)e[g(x)], p( ) b( ;,p) = E { [( X)p σ 2 w(x)] g(x) } + (p µ)e[g(x)]

7 ON POINTWISE BINOMIAL APPROXIMATION E{ ( X)p σ 2 w(x) g(x) } + p µ E g(x). Hece, by usig Lemma 2.1, the theorem is proved. The followig corollary is a cosequece of Theorem 3.1. Corollary 3.1. If ( x)p σ 2 w(x) / < 0 for every x S(x), the 1. For = 0, ad, if p = µ, the p(0) b(0;,p) 1 q p p(0) b(0;,p) 1 q p { ( µ)p σ 2 + p µ } (3.6) µq σ 2. (3.7) 2. For {1,...,}, { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } { ( µ)p σ 2 } + p µ (3.8) ad, if p = µ, the { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } µq σ 2. (3.9) 4. Applicatios This sectio, we apply the result i Theorem 3.1 to approximate some distributios such as hypergeometric, egative hypergeometric ad Pólya distributios Applicatio to Hypergeometric Distributio Suppose a radom sample of size is draw without replacemet from a fiite populatio cotaiig N elemets of two types of which m are of type I ad N m are of type II. Let X be the umber of type I elemets i the sample.

8 64 K. Teerapabolar, P. Wogkasem The X has a hypergeometric distributio with parameters N, ad m ad has probability fuctio as follows: ( m N m ) p(x) = x)( x ( N, x = 0,1,...,mi{,m}. ) Here, its mea ad variace are µ = m N ad σ2 = m(n )(N m), respectively. N 2 (N 1) It is well-kow that the hypergeometric distributio ca be approximated by the biomial distributio. For this applicatio, we give a result of the biomial approximatio to the hypergeometric distributio i terms of the poit metric p( ) b( ;,p), where {0,1,...,mi{,m}}. Followig the relatio (2.2), we have w(x) = ( x)(m x). If mi{,m} =, Nσ 2 we put p = m N i Theorem 3.1, the ( x)p σ2 w(x) = ( x)m N ( x)(m x) N 0 for all 0 x. By applyig Corollary 3.1, a result of this approximatio ca be expressed as the followig. Corollary 4.1. For p = m N, p( ) b( ;,p) { (1 q )q( 1) N 1 { } mi 1 p, 1 p+1 q +1 ( 1)p (+1)p N 1 if 1. Similarly, if mi{,m} = m, we replace ad p i Theorem 3.1 by m ad N, respectively, ad usig Corollary 3.1, we ca obtai a aother result of this approximatio as the followig corollary. Corollary 4.2. If p = N, the we have p( ) b( ;m,p) { (1 q m )q(m 1) { N 1 } mi 1 p m, 1 pm+1 q m+1 (m 1)mp (m+1)p N 1 if 1 m. Remark 4.1. It should be oted that each result i Corollaries 4.1 ad 4.2 gives a good biomial approximatio if N is large ad ad m are small, or are small. N ad m N 4.2. Applicatio to Negative Hypergeometric Distributio Let us cosider the process of samplig without replacemet as metioed i previous subsectio. If elemets i a radom sample are draw without replacemet from this populatio util the umber of types II elemets reaches

9 ON POINTWISE BINOMIAL APPROXIMATION a fixed positive iteger r ad let X be the umber of types I elemets i the sample. The X has a egative hypergeometric distributio with parameters N,m ad r, ad its probability fuctio ca be expressed as ) p(x) = ( r+x 1 )( N r x x m x ( N m), x = 0,1,...,m, where r {1,...,N m} ad µ = rm N m+1 ad σ2 = rm(n m r+1)(n+1) are the (N m+1) 2 (N m+2) mea ad variace of X, respectively. It is observed that, if N,r such that r N m+1 teds to a costat θ, the a egative hypergeometric distributio with parameters N, m ad r coverges to a biomial distributio with parameters m ad θ. Therefore, we ca also use the biomial probability fuctio to approximate the egative hypergeometric probability fuctio by usig certai coditios of this covergece. Usig the relatio (2.2), the w(x) = (r+x)(m x). Thus, replacig by m (N m+1)σ 2 r ad p by N m+1 i Theorem 3.1, we have (m x)p σ2 w(x) = r(m x) N m+1 (r+x)(m x) N m+1 0 for all 0 x m. By Corollary 3.1, the followig corollary is obtaied. Corollary 4.3. For r {1,...,N m}, if p = p( ) b( ;m,p) { (1 q m )q(m 1) r N m+1, the N m+2 { } mi 1 p m, 1 pm+1 q m+1 (m 1)mp (m+1)p N m+2 if 1 m Applicatio to Pólya Distributio Suppose that a sigle ur cotai r red ad N r black balls. Draw a ball at radom, ote the color, ad retur it ito the ur together with a additioal ball of the same color. Repeat this way for m draws. Let X be the umber of red balls take out i the m drawigs, the the distributio of X is a Pólya distributio with parameters N, m ad r. The probability fuctio of X is give by ) p ( x) = ( r+x 1 )( N r+m x 1 x m x ( N+m 1 m ), x = 0,1,...,m ad the mea ad variace of X are µ = rm N ad σ2 = rm(n+m)(n r), respectively. N 2 (N+1) Usig the relatio (2.2), we the obtai w(x) = (r+x)(m x). Settig = m Nσ 2 ad p = r N i Theorem 3.1, it follows that (m x)p σ2 w(x) = (m x)x N 0

10 66 K. Teerapabolar, P. Wogkasem for all 0 x m. The followig corollary is directly obtaied from Corollary 3.1. Corollary 4.4. If p = r N, the we have the followig. p( ) b( ;m,p) { (1 q m )q(m 1) { N+1 } mi 1 p m, 1 pm+1 q m+1 (m 1)mp (m+1)p N+1 if 1 m. Remark 4.2. Each result i Corollaries 4.3 ad 4.4 yields a good approximatio as N is large ad m is small, or m N is small. Ackowledgmets The authors would like to thak Faculty of Sciece, Burapha Uiversity, for fiacial support to do this research. Refereces [1] A.D. Barbour, L. Holst, S. Jaso, Poisso Approximatio, Oxford Studies i Probability 2, Claredo Press, Oxford (1992). [2] T. Cacoullos, V. Papathaasio, Characterizatio of distributios by variace bouds, Statist. Probab. Lett., 7 (1989), [3] W. Ehm, Biomial approximatio to the Poisso biomial distributio, Statist. Probab. Lett., 11 (1991), [4] M. Majserowska, A ote o Poisso approximatio by w-fuctios, Appl. Math., 25 (1998), [5] Y.T. Soo Spario, Biomial approximatio for depedet idicators, Statist. Siica, 6 (1996), [6] C.M. Stei, Approximate Computatio of Expectatios, IMS, Hayward Califoria (1986). [7] K. Teerapabolar, A boud o the biomial approximatio to the beta biomial distributio, It. Math. Forum, 3 (2008), [8] P. Wogkasem, K. Teerapabolar, R. Gulasirima, O approximatig a geeralized biomial by biomial ad Poisso distributios, Iterat. J. Statist. Systems, 3 (2008),

Songklanakarin Journal of Science and Technology SJST R1 Teerapabolarn

Songklanakarin Journal of Science and Technology SJST R1 Teerapabolarn Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar A No-uiform Boud o Biomial Aroimatio to the Beta Biomial Cumulative Distributio Fuctio Joural: Soglaaari Joural of Sciece ad Techology For Review

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

More information

Information Theory and Statistics Lecture 4: Lempel-Ziv code

Information Theory and Statistics Lecture 4: Lempel-Ziv code Iformatio Theory ad Statistics Lecture 4: Lempel-Ziv code Łukasz Dębowski ldebowsk@ipipa.waw.pl Ph. D. Programme 203/204 Etropy rate is the limitig compressio rate Theorem For a statioary process (X i)

More information

γn 1 (1 e γ } min min

γn 1 (1 e γ } min min Hug ad Giag SprigerPlus 20165:79 DOI 101186/s40064-016-1710-y RESEARCH O bouds i Poisso approximatio for distributios of idepedet egative biomial distributed radom variables Tra Loc Hug * ad Le Truog Giag

More information

Element sampling: Part 2

Element sampling: Part 2 Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

More information

Large holes in quasi-random graphs

Large holes in quasi-random graphs Large holes i quasi-radom graphs Joaa Polcy Departmet of Discrete Mathematics Adam Mickiewicz Uiversity Pozań, Polad joaska@amuedupl Submitted: Nov 23, 2006; Accepted: Apr 10, 2008; Published: Apr 18,

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Basics of Probability Theory (for Theory of Computation courses)

Basics of Probability Theory (for Theory of Computation courses) Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.

More information

SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE (1 + 1/n) n

SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE (1 + 1/n) n SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE + / NECDET BATIR Abstract. Several ew ad sharp iequalities ivolvig the costat e ad the sequece + / are proved.. INTRODUCTION The costat e or

More information

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece

More information

4. Partial Sums and the Central Limit Theorem

4. Partial Sums and the Central Limit Theorem 1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems

More information

Application to Random Graphs

Application to Random Graphs A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let

More information

Convergence of random variables. (telegram style notes) P.J.C. Spreij

Convergence of random variables. (telegram style notes) P.J.C. Spreij Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space

More information

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314

More information

Chapter 6 Infinite Series

Chapter 6 Infinite Series Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

More information

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial. Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable

More information

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A. Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)

More information

B Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets

B Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets B671-672 Supplemetal otes 2 Hypergeometric, Biomial, Poisso ad Multiomial Radom Variables ad Borel Sets 1 Biomial Approximatio to the Hypergeometric Recall that the Hypergeometric istributio is fx = x

More information

Self-normalized deviation inequalities with application to t-statistic

Self-normalized deviation inequalities with application to t-statistic Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric

More information

Binomial approximation to the Markov binomial distribution

Binomial approximation to the Markov binomial distribution Biomial approximatio to the Markov biomial distributio V. Čekaavičius ad B. Roos Vilius Uiversity ad Uiversity of Hamburg 2d November 2006 Abstract The Markov biomial distributio is approximated by the

More information

Asymptotic distribution of products of sums of independent random variables

Asymptotic distribution of products of sums of independent random variables Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Chapter 5. Inequalities. 5.1 The Markov and Chebyshev inequalities

Chapter 5. Inequalities. 5.1 The Markov and Chebyshev inequalities Chapter 5 Iequalities 5.1 The Markov ad Chebyshev iequalities As you have probably see o today s frot page: every perso i the upper teth percetile ears at least 1 times more tha the average salary. I other

More information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor IJITE Vol Issue-, (November 4) ISSN: 3-776 ATTRACTIVITY OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Guagfeg Liu School of Zhagjiagag Jiagsu Uiversit of Sciece ad Techolog, Zhagjiagag, Jiagsu 56,PR

More information

Sieve Estimators: Consistency and Rates of Convergence

Sieve Estimators: Consistency and Rates of Convergence EECS 598: Statistical Learig Theory, Witer 2014 Topic 6 Sieve Estimators: Cosistecy ad Rates of Covergece Lecturer: Clayto Scott Scribe: Julia Katz-Samuels, Brado Oselio, Pi-Yu Che Disclaimer: These otes

More information

An Introduction to Randomized Algorithms

An Introduction to Randomized Algorithms A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis

More information

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

Exponential Families and Bayesian Inference

Exponential Families and Bayesian Inference Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where

More information

Entropy and Ergodic Theory Lecture 5: Joint typicality and conditional AEP

Entropy and Ergodic Theory Lecture 5: Joint typicality and conditional AEP Etropy ad Ergodic Theory Lecture 5: Joit typicality ad coditioal AEP 1 Notatio: from RVs back to distributios Let (Ω, F, P) be a probability space, ad let X ad Y be A- ad B-valued discrete RVs, respectively.

More information

Randomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)

Randomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018) Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES. 1. Introduction Basic hypergeometric series (cf. [GR]) with the base q is defined by

ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES. 1. Introduction Basic hypergeometric series (cf. [GR]) with the base q is defined by ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES TOSHIO OSHIMA Abstract. We examie the covergece of q-hypergeometric series whe q =. We give a coditio so that the radius of the covergece is positive ad get

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak

More information

Estimation for Complete Data

Estimation for Complete Data Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

Central limit theorem and almost sure central limit theorem for the product of some partial sums

Central limit theorem and almost sure central limit theorem for the product of some partial sums Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp. 289 294. Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics

More information

Berry-Esseen bounds for self-normalized martingales

Berry-Esseen bounds for self-normalized martingales Berry-Essee bouds for self-ormalized martigales Xiequa Fa a, Qi-Ma Shao b a Ceter for Applied Mathematics, Tiaji Uiversity, Tiaji 30007, Chia b Departmet of Statistics, The Chiese Uiversity of Hog Kog,

More information

The Boolean Ring of Intervals

The Boolean Ring of Intervals MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,

More information

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f. Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,

More information

Properties of Fuzzy Length on Fuzzy Set

Properties of Fuzzy Length on Fuzzy Set Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,

More information

Solutions to HW Assignment 1

Solutions to HW Assignment 1 Solutios to HW: 1 Course: Theory of Probability II Page: 1 of 6 Uiversity of Texas at Austi Solutios to HW Assigmet 1 Problem 1.1. Let Ω, F, {F } 0, P) be a filtered probability space ad T a stoppig time.

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

Rates of Convergence for Quicksort

Rates of Convergence for Quicksort Rates of Covergece for Quicksort Ralph Neiiger School of Computer Sciece McGill Uiversity 480 Uiversity Street Motreal, HA 2K6 Caada Ludger Rüschedorf Istitut für Mathematische Stochastik Uiversität Freiburg

More information

Poisson approximations

Poisson approximations The Bi, p) ca be thought of as the distributio of a sum of idepedet idicator radom variables X +...+ X, with {X i = } deotig a head o the ith toss of a coi. The ormal approximatio to the Biomial works

More information

0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =

0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) = PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several well-ow discrete probability distributios ad study some of their properties. Some of these distributios, lie

More information

EE 4TM4: Digital Communications II Probability Theory

EE 4TM4: Digital Communications II Probability Theory 1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a real-valued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair

More information

A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca

A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity

More information

Law of the sum of Bernoulli random variables

Law of the sum of Bernoulli random variables Law of the sum of Beroulli radom variables Nicolas Chevallier Uiversité de Haute Alsace, 4, rue des frères Lumière 68093 Mulhouse icolas.chevallier@uha.fr December 006 Abstract Let be the set of all possible

More information

Lecture 8: Convergence of transformations and law of large numbers

Lecture 8: Convergence of transformations and law of large numbers Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges

More information

Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].

Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)]. Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x

More information

Lecture 2. The Lovász Local Lemma

Lecture 2. The Lovász Local Lemma Staford Uiversity Sprig 208 Math 233A: No-costructive methods i combiatorics Istructor: Ja Vodrák Lecture date: Jauary 0, 208 Origial scribe: Apoorva Khare Lecture 2. The Lovász Local Lemma 2. Itroductio

More information

Lecture 7: Properties of Random Samples

Lecture 7: Properties of Random Samples Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ

More information

APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS

APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS Hacettepe Joural of Mathematics ad Statistics Volume 32 (2003), 1 5 APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS E. İbili Received 27/06/2002 : Accepted 17/03/2003 Abstract The weighted approximatio

More information

Detailed proofs of Propositions 3.1 and 3.2

Detailed proofs of Propositions 3.1 and 3.2 Detailed proofs of Propositios 3. ad 3. Proof of Propositio 3. NB: itegratio sets are geerally omitted for itegrals defied over a uit hypercube [0, s with ay s d. We first give four lemmas. The proof of

More information

f(x i ; ) L(x; p) = i=1 To estimate the value of that maximizes L or equivalently ln L we will set =0, for i =1, 2,...,m p x i (1 p) 1 x i i=1

f(x i ; ) L(x; p) = i=1 To estimate the value of that maximizes L or equivalently ln L we will set =0, for i =1, 2,...,m p x i (1 p) 1 x i i=1 Parameter Estimatio Samples from a probability distributio F () are: [,,..., ] T.Theprobabilitydistributio has a parameter vector [,,..., m ] T. Estimator: Statistic used to estimate ukow. Estimate: Observed

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

On Random Line Segments in the Unit Square

On Random Line Segments in the Unit Square O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,

More information

Disjoint Systems. Abstract

Disjoint Systems. Abstract Disjoit Systems Noga Alo ad Bey Sudaov Departmet of Mathematics Raymod ad Beverly Sacler Faculty of Exact Scieces Tel Aviv Uiversity, Tel Aviv, Israel Abstract A disjoit system of type (,,, ) is a collectio

More information

Chapter 6 Principles of Data Reduction

Chapter 6 Principles of Data Reduction Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a

More information

arxiv: v1 [math.fa] 3 Apr 2016

arxiv: v1 [math.fa] 3 Apr 2016 Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert

More information

Math 155 (Lecture 3)

Math 155 (Lecture 3) Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4. 4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad

More information

Axioms of Measure Theory

Axioms of Measure Theory MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that

More information

SDS 321: Introduction to Probability and Statistics

SDS 321: Introduction to Probability and Statistics SDS 321: Itroductio to Probability ad Statistics Lecture 23: Cotiuous radom variables- Iequalities, CLT Puramrita Sarkar Departmet of Statistics ad Data Sciece The Uiversity of Texas at Austi www.cs.cmu.edu/

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

The log-behavior of n p(n) and n p(n)/n

The log-behavior of n p(n) and n p(n)/n Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity

More information

The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution

The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters

More information

k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction

k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet

More information

Lecture 33: Bootstrap

Lecture 33: Bootstrap Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece

More information

COMMON FIXED POINT THEOREMS VIA w-distance

COMMON FIXED POINT THEOREMS VIA w-distance Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3, Pages 182-189 COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA

More information

PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION

PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu

More information

Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable

Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya

More information

CS 171 Lecture Outline October 09, 2008

CS 171 Lecture Outline October 09, 2008 CS 171 Lecture Outlie October 09, 2008 The followig theorem comes very hady whe calculatig the expectatio of a radom variable that takes o o-egative iteger values. Theorem: Let Y be a radom variable that

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

A constructive analysis of convex-valued demand correspondence for weakly uniformly rotund and monotonic preference

A constructive analysis of convex-valued demand correspondence for weakly uniformly rotund and monotonic preference MPRA Muich Persoal RePEc Archive A costructive aalysis of covex-valued demad correspodece for weakly uiformly rotud ad mootoic preferece Yasuhito Taaka ad Atsuhiro Satoh. May 04 Olie at http://mpra.ub.ui-mueche.de/55889/

More information

Lecture 12: September 27

Lecture 12: September 27 36-705: Itermediate Statistics Fall 207 Lecturer: Siva Balakrisha Lecture 2: September 27 Today we will discuss sufficiecy i more detail ad the begi to discuss some geeral strategies for costructig estimators.

More information

Math 104: Homework 2 solutions

Math 104: Homework 2 solutions Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does

More information

M17 MAT25-21 HOMEWORK 5 SOLUTIONS

M17 MAT25-21 HOMEWORK 5 SOLUTIONS M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series

More information

RANDOM WALKS ON THE TORUS WITH SEVERAL GENERATORS

RANDOM WALKS ON THE TORUS WITH SEVERAL GENERATORS RANDOM WALKS ON THE TORUS WITH SEVERAL GENERATORS TIMOTHY PRESCOTT AND FRANCIS EDWARD SU Abstract. Give vectors { α i } [0, 1)d, cosider a radom walk o the d- dimesioal torus T d = R d /Z d geerated by

More information

Discrete probability distributions

Discrete probability distributions Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop

More information

Rademacher Complexity

Rademacher Complexity EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for

More information

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n. Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

More information

Enumerative & Asymptotic Combinatorics

Enumerative & Asymptotic Combinatorics C50 Eumerative & Asymptotic Combiatorics Stirlig ad Lagrage Sprig 2003 This sectio of the otes cotais proofs of Stirlig s formula ad the Lagrage Iversio Formula. Stirlig s formula Theorem 1 (Stirlig s

More information

ST5215: Advanced Statistical Theory

ST5215: Advanced Statistical Theory ST525: Advaced Statistical Theory Departmet of Statistics & Applied Probability Tuesday, September 7, 2 ST525: Advaced Statistical Theory Lecture : The law of large umbers The Law of Large Numbers The

More information

Approximation theorems for localized szász Mirakjan operators

Approximation theorems for localized szász Mirakjan operators Joural of Approximatio Theory 152 (2008) 125 134 www.elsevier.com/locate/jat Approximatio theorems for localized szász Miraja operators Lise Xie a,,1, Tigfa Xie b a Departmet of Mathematics, Lishui Uiversity,

More information

Monte Carlo Integration

Monte Carlo Integration Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

More information

MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES

MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES Commu Korea Math Soc 26 20, No, pp 5 6 DOI 0434/CKMS20265 MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES Wag Xueju, Hu Shuhe, Li Xiaoqi, ad Yag Wezhi Abstract Let {X, } be a sequece

More information

Lecture 2: Concentration Bounds

Lecture 2: Concentration Bounds CSE 52: Desig ad Aalysis of Algorithms I Sprig 206 Lecture 2: Cocetratio Bouds Lecturer: Shaya Oveis Ghara March 30th Scribe: Syuzaa Sargsya Disclaimer: These otes have ot bee subjected to the usual scrutiy

More information

New Inequalities For Convex Sequences With Applications

New Inequalities For Convex Sequences With Applications It. J. Ope Problems Comput. Math., Vol. 5, No. 3, September, 0 ISSN 074-87; Copyright c ICSRS Publicatio, 0 www.i-csrs.org New Iequalities For Covex Sequeces With Applicatios Zielaâbidie Latreuch ad Beharrat

More information

2.1. Convergence in distribution and characteristic functions.

2.1. Convergence in distribution and characteristic functions. 3 Chapter 2. Cetral Limit Theorem. Cetral limit theorem, or DeMoivre-Laplace Theorem, which also implies the wea law of large umbers, is the most importat theorem i probability theory ad statistics. For

More information

Random Matrices with Blocks of Intermediate Scale Strongly Correlated Band Matrices

Random Matrices with Blocks of Intermediate Scale Strongly Correlated Band Matrices Radom Matrices with Blocks of Itermediate Scale Strogly Correlated Bad Matrices Jiayi Tog Advisor: Dr. Todd Kemp May 30, 07 Departmet of Mathematics Uiversity of Califoria, Sa Diego Cotets Itroductio Notatio

More information

Mi-Hwa Ko and Tae-Sung Kim

Mi-Hwa Ko and Tae-Sung Kim J. Korea Math. Soc. 42 2005), No. 5, pp. 949 957 ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF NEGATIVELY ORTHANT DEPENDENT RANDOM VARIABLES Mi-Hwa Ko ad Tae-Sug Kim Abstract. For weighted sum of a sequece

More information

Advanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology

Advanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4

More information

arxiv: v1 [cs.sc] 2 Jan 2018

arxiv: v1 [cs.sc] 2 Jan 2018 Computig the Iverse Melli Trasform of Holoomic Sequeces usig Kovacic s Algorithm arxiv:8.9v [cs.sc] 2 Ja 28 Research Istitute for Symbolic Computatio RISC) Johaes Kepler Uiversity Liz, Alteberger Straße

More information

A 2nTH ORDER LINEAR DIFFERENCE EQUATION

A 2nTH ORDER LINEAR DIFFERENCE EQUATION A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy

More information