An Exceptional Generalization of the Poisson Distribution
|
|
- Ezra Cobb
- 5 years ago
- Views:
Transcription
1 Open Journal of Saisics, 1,, hp://dx.doi.org/1.436/ojs Published Online July 1 (hp:// An Excepional Generalizaion of he Poisson Disribuion Per-Erik Hagmark Deparmen of Mechanics and Design, Tampere Universiy of Technology, Tampere, Finland per-erik.hagmark@u.fi Received May 4, 1; revised June 1, 1; acceped June 3, 1 ABSTRACT A new wo-parameer coun disribuion is derived saring wih probabilisic argumens around he gamma funcion and he digamma funcion. This model is a generalizaion of he Poisson model wih a noeworhy assormen of qualiies. For example, he mean is he main model parameer; any possible non-rivial variance or zero probabiliy can be aained by changing he oher model parameer; and all disribuions are visually naural-shaped. Thus, exac modeling o any degree of over/under-dispersion or zero-inflaion/deflaion is possible. Keywords: Coun Daa; Gamma Funcion; Poisson Generalizaion; Discreizaion; Modeling; Over/Under-Dispersion; Zero-Inflaion/Deflaion 1. Inroducion and he Main Resul In coun daa modeling he Poisson disribuion is usually he firs opion, bu real daa can indicae a variey of discrepancies. These can be genuine feaures or secondary consequences of e.g. censoring, clusering, approximaions or correlaions. Specifically, he Poisson model has no dispersion flexibiliy because he mean deermines he variance and he zero probabiliy, σ = μ, p = e μ, while he real daa can display over or under- dispersion, σ μ, or zero-inflaion or deflaion, p e μ [1]. Such siuaions are usually handled e.g. by randomizing he Poisson mean, by mixures, by adding a new parameer, by reweighing he Poisson poin probabiliies, or via generalizing he exponenial incremens in he homogeneous Poisson process [-5]. Our approach will be differen. We recall an elemenary fac. The mean-deviaion pair (μ, σ) of a non-binary coun variable (non-negaive ineger-valued random variable) always saisfies he inequaliy 1, (1) where [μ] is he larges ineger no exceeding μ. Thus, we will say ha a coun model (parameerized coun variable) has full dispersion flexibiliy if every posiive soluion (μ, σ) of he inequaliy (1) is he mean-deviaion pair for some parameer values. In [6] we called for a mahemaically unified coun model N(μ, β) wih wo independen parameers, µ >, β >, and he following properies: 1) Comforable parameerizaion: E(N(μ, β)) = μ, for all μ and β. ) Generalizaion of he Poisson model: For β = 1, n Pr N μ,1 n e n!, n =, 1,. 3) Full dispersion flexibiliy: If he numbers μ > and σ > saisfy inequaliy (1), hen here is a β such ha Var N,. The soluion o be presened in his paper obeys he following cumulaive probabiliies: Pr N μ, β n n n 1 ng, ng, 1 () n1 n1 n1 G, n1 g, 1, where g(, x) and G(, x) are he one-parameer gamma probabiliy and cumulaive disribuion funcions, respecively, wih parameer x and variable (Secion ). We begin wih he derivaion of fundamenal inequaliies in Secion. These inequaliies lead o a cumulaive disribuion H(x, μ), where he parameer μ > is he mean. Then he inserion of a new independen parameer β > provides an exended cumulaive disribuion H(x/β, μ/β) and he relaed non-negaive wo-parameer random variable X(μ, β), where μ is sill he mean. Now he proclaimed coun model N(μ, β) is defined as a mean-preserving discreizaion of X(μ, β), and he above properies 1), ), 3) are proved. Thereafer he mos immediae applicaions are given; namely, exac modeling of over/ under-dispersion or zero-inflaion/deflaion o any possible degree. In he las secion, we propose moives for furher research, and we compare N(μ, β) wih well-esablished Poisson generalizaions. Copyrigh 1 SciRes.
2 314 P.-E. HAGMARK. Derivaion of Two Inequaliies We sar wih noaion: Gamma funcion Г(x) as Euler s second inegral, digamma funcion Ψ(x), some relaed funcions and immediae inerrelaions; x x1 : e d, x, x: x x, x 1, : e g x x,, G, x: g s, x d s, ax (, ): gx, gx, ln x, x bx, : ax, gx, ln x x, x Ax, : Gx, asx, )d s, x x B, x: A, x b s x, d s. There is a nice probabilisic perspecive on he gamma funcion: If he random variable T has a gamma densiy g(, x), hen E(ln(T)) = Ψ(x) and Var(ln(T)) = dψ(x)/dx [7]. In erms of our noaion above, hese simple observaions can be wrien in he form lim Ax,, lim Bx,, x. (3) Addiional work leads o a sronger resul, Ax, d 1, Bx, d, x. (4) Namely, inegraion by pars, he funcional equaions x 1 xx, gx, xgx, 1, formula (3), and l Hospial s rule allow us o wrie Ax, d Ax gx, 1ln x x1 x x 1, x d lim Bx, gx, ln x x g, x 1 ln x x x1x lim (, ), ln x d x g x B, x d x x x d d. Nex we derive wo fundamenal inequaliies. For every fixed x >, he funcion a(, x) has exacly one roo x e, and i is increasing here. This and (3, lef side) imply Ax,,, x. 1 Ax, d, x,. (5) Now, aking ino accoun (5) and (4, lef side), we obain he firs inequaliy, 1 (6) Furher, for every fixed x >, he funcion b(, x) has x x exacly wo roos, e x e B, x d, x,. x, and i is decreasing a and increasing a 1. From his one can conclude ha B(, x) has, for every x >, a posiive local maximum a and, because of (3, righ side), a negaive local minimum a 1. Considering (4, righ side) oo, we finally arrive a he second inequaliy 3. A Mean-Preserving Discreizaion We will also need a cerain discreizaion procedure: If X is a non-negaive random variable wih cumulaive disribuion F(x), he discreizaion of X is a coun variable N wih cumulaive probabiliies equal o he mean F(x) on he inerval (n, n + 1), i.e. n1 Pr N n : F x d x, n,1, n We shorly quoe he basic properies from [6]: The mean and he variance of N exis (are finie) if and only if he mean and he variance of X exis, and in ha case N X X N X X (7) (8) E E, (9) Var Var Var min E, A Generalizaion of he Poisson Model (1) In our consrucion of a new generalizaion of he Poisson model, he following one-parameer funcion will be he cenral ingredien: H x, : 1 G, xd. (11) x Recalling (5) and he noaion A(, x) = G(, x)/ x from Secion, we derive 1 Hx, d x A, xdx d. (1) In (1) we firs changed he inegraion order (as he inegrand is posiive) and hen employed he limis G x G, : lim x, 1, (13a) G x G, : lim, (13b) x. Copyrigh 1 SciRes.
3 P.-E. HAGMARK 315 The limis (13) follow from Chebyshev s inequaliy and he simple fac ha he parameer x of he one-parameer gamma densiy g(, x) equals he mean and he variance. By employing he inequaliies (6) and (7), we have < H(x, μ) < 1 and H(x, μ)/ x >. Hence, H(x, μ) is a cumulaive probabiliy disribuion wih mean μ (1) and zero probabiliy H, : lim x Hx,. We proceed by adding an independen parameer β >, so defining a wo-parameer cumulaive disribuion, x Fx,, : H,, x. (14) Now, le X(μ, β) be he non-negaive random variable deermined by F(x, μ, β), and le N(μ, β) be he discreizaion of X(μ, β), according o Secion 3. We form an inegral funcion of (14) and ge he cumulaive probabiliies of N(μ, β) using (8): I x,, : F x,, dx x x Gs, ds x x G, d,,, : Pr, ) In In P n N n 1,,,, (15) (16) n n1 1 G, G, d The pair X(μ, β) and N(μ, β) is illusraed in Figure 1. Proof of Properies 1) and ), Secion 1. By considering (9, 1, 14) one can see ha he mean does no change during he process from H(x, μ) o N(μ, β): x E N μβ, E X μβ, 1 H, dx β β (17) E X,1, proving Propery 1). Nex, we fix β = 1 in (16) and employ he ideniies G(, x) G(, x + 1) = g(, x + 1) and G(, ) = 1 (13a). Now Pr{N(μ, 1) n} = 1 G(μ, n + 1), so he poin probabiliies are Pr{N(μ, 1) = n} = G(μ, n) n G(μ, n + 1) = e n!, n =, 1,. This means ha he sub-model N(μ, 1) is he Poisson model, so Propery ) holds rue (see case β = 1 in Figure 1). 5. Full Dispersion Flexibiliy Propery 3), Secion 1, remains o be proved. Given any posiive pair (μ, σ) saisfying 1, we have o prove ha here is a β > such ha Var(N(μ, β)) = σ. Figure is an illusraion. Firs, one obains an upper bound for he variance of X(μ, β) by employing Properies 1) and ), (1, lef side) and rouines: Var X, x 1 H x, dx E X,1 Var X,1 N Var,1. (18) Then (18) and (1, righ side) imply Var(N(μ, β)) <. Afer noing ha Var(N(μ, β)) is a coninuous funcion of β (for fixed μ) and recalling inequaliy (1), i is enough o prove he following limis: lim Var N, 1, (19) lim Var N, () Figure 1. Cumulaive disribuions of X(μ, β) and N(μ, β), for μ = 3. and β = 1,.6, 4,.1. Copyrigh 1 SciRes.
4 316 P.-E. HAGMARK Figure. The variance Var(N(μ, β)) as a funcion of β, for μ = 3. and μ =.7. Poisson poin (β = 1, σ = μ); lower bound = μ μ 1 μ+ μ. σmin Proof of (19). From (18) i follows ha Var(X(μ, β)) ends o zero as β. This means ha X(μ, β) approaches he consan µ (in disribuion). This again means ha he discreizaion N(μ, β) approaches μ if his is an ineger, and oherwise a binary coun variable wih he values [μ] and [μ]+1; see [6]. In boh cases he limi of Var(N(μ, β)) obeys (19). Proof of (). Definiion (11) and parial inegraion yield he ideniy M x 1 H x, dx M M G, Md G x x, d d. The firs erm on he righ side vanishes when M, since MG(, M) M /Г(M). Now by changing he inegraion order in he laer erm, one obains where E X,1 x 1 H x, dx L s ds d, s x1 L s : g s, x d xe s x dx. (1) Then, by using (1) and par of (18), and changing inegraion variable, z = β, one arrives a X X,1 E, E z z Lsdsd z. z x 1 Furher, he inequaliy s 1 x1 ln s >, yields a lower bound for L(s): 1 L s g s, x dxe x s C Dln s, x 1 C dx, D dx. x (), s >, x This means ha L(s) ends o as s, and so he average of L in he inerval (, z/β) approaches as β (). Thereby, E(X(μ, β) ) grows o, so (17) and (1, lef side) complee he proof of (). 6. Compuing and Applicaions When working wih N(μ, β), he following numbers are useful: K n, n : G, d n n ng, ng, 1, n,1, (3) The laer faser version follows from parial inegraion and he ideniies G(, x) G(, x + 1) = g(, x + 1), G(, ) = 1 (13a). Noe also ha mos mahemaical sofware offers fas compuaion of G(, x). Employing (3) in (16), basic formulas can be wrien in he following form: N n Kn Kn 1 Pr, ) 1,,, E N, n1 k k k k n n n Kn 1 1,, N Kn n1 4) (5) Var,,. (6) We consider exac modeling of coun variables. (For numerical examples, see Table 1). Applicaion 1. Generally, a non-binary coun variable wih desired mean μ and variance σ exiss if and only if 1. (7) In ha case N(μ, β) always provides a soluion. Indeed, because of full dispersion flexibiliy, Propery 3), here Copyrigh 1 SciRes.
5 P.-E. HAGMARK 317 Table 1. Under/over-dispersion and zero-deflaion/inflaion. Phenomenon General range Numerical example Soluion Under-dispersion (μ [μ])(1 μ + [μ]) < σ < μ μ = 3. σ =.4 β =.753 Poisson σ = μ (equi-dispersion) μ = 3. σ = 3. β = 1 Over-dispersion μ < σ < μ = 3. σ = 4.5 β = Zero-deflaion max{,1 μ}< p < e μ μ = 3. p =.1 β =.56 Poisson p = e μ μ = 3. p = β = 1 Zero-inflaion e μ < p < 1 μ = 3. p =.15 β =.949 is a β > such ha Var(N(μ, β)) = σ (6). Applicaion. Likewise, a non-binary coun variable wih desired mean μ and zero probabiliy p exiss if and only if p max, 1 1. (8) Again N(μ, β) provides a soluion. Argumens like hose in Secion 5 would show ha here is a β > such ha Pr N, ) = p (4, n = ). Applicaion 3. Suppose here is a real non-censored random sample available of he unknown non-binary coun variable o be modeled. Le ˆ be he sample mean, ˆ he sandard variance and ˆp he zero fracion. I is easy o prove ha hese UMVU esimaes also mee (7, 8). Thus, here is a β 1 ha saisfies ˆ and a β ha saisfies ˆp (boh exacly), bu of course, usually 1. Imporance weighing provides a compromise β and an approximae soluion N ˆ,. 7. Furher Research and Discussion Addiional work is needed o enlarge he applicabiliy of N(μ, β). The compuaional behavior of he cenral formulas 3-6 should be furher explored, and ools for sochasic simulaion and saisical inference should be developed. We pu forward wo concree problems. Problem 1. Numerical experimenaion indicaes ha he numbers K n (3, n 1) increase wih β (K = μ). If his is rue, all momens (5, k ) increase wih β, so he ieraion of β in he applicaions in Secion 6 can be made faser. Problem. Find an algorihm for generaion of random variaes from N(μ, β). The alias mehod [8] can of course be used for runcaed versions, bu a ailor-made mehod would be welcome. Acually, a generaion mehod for X(μ, β) would be enough since, according o [6], his can immediaely be ransformed o he discreizaion N(μ, β). Finally, we reurn o he main qualiies of N(μ, β). As menioned, he finie mean-deviaion pair (μ, σ) of any non-binary coun variable saisfies inequaliy (1), i.e. σ > 1. Conversely, if (μ, σ) is a posiive soluion of (1), hen i is he mean-deviaion pair of a non-binary coun variable; and as we have shown, here is always an N(μ, β) wih his mean-deviaion pair. Since he mean is an original model parameer of N(μ, β), only β needs o be solved from he equaion Var(N(μ, β)) = σ. We have called his feaure full dispersion flexibiliy, because i enables exac modeling for he firs wo momens, or for mean and zero probabiliy. Full dispersion flexibiliy seems o be very rare even among well-esablished Poisson generalizaions. The generalizaion of Consul and Jain [], he negaive binomial [3], he COM-Poisson disribuion [4] and many ohers have severe shorcomings in dispersion flexibiliy, and also parly bad-shaped disribuion funcions. A posiive excepion is he General Poisson Law [5]. However, here he mean is no a model parameer, so, if a cerain pair (μ, σ) is waned, he original parameers mus be solved simulaneously from wo equaions, which boh include laborious infinie series. Also noe ha he invarians (4) and (5), he inequaliies (6) and (7), and he disribuion (11) comprise, as such, a conribuion o probabilisic reamen of he gamma funcion. REFERENCES [1] J. Casillo and M. Perez-Casany, Over-Dispersed and Under-Dispersed Poisson Generalizaions, Journal of Saisical Planning and Inference, Vol. 134, No., 5, pp doi:1.116/j.jspi [] P. C. Consul and G. C. Jain, A Generalizaion of he Poisson Disribuion, Technomerics, Vol. 15, No. 4, 1973, pp doi:1.37/ [3] N. L. Johnson, S. Koz and A. W. Kemp, Univariae Discree Disribuions, nd Ediion, John Wiley & Sons, New York, 199. [4] R. W. Conway and W. L. Maxwell, A Queuing Model wih Sae Dependen Service Raes, Journal of Indusrial Engineering, Vol. 1, 196, pp [5] G. Morla, Sur Une Généralisaion de la loi de Poisson, Compes Redus, Vol. 35, 195, pp [6] P.-E. Hagmark, On Consrucion and Simulaion of Coun Copyrigh 1 SciRes.
6 318 P.-E. HAGMARK Daa Models, Mahemaics and Compuers in Simulaion, Vol. 77, No. 1, 8, pp doi:1.116/j.macom [7] L. Gordon, A Sochasic Approach o he Gamma Funcion, The American Mahemaical Monhly, Vol. 11, No. 9, 1994, pp [8] A. J. Walker, An Efficien Mehod for Generaing Discree Random Variables wih General Disribuions, ACM Transacions on Mahemaical Sofware, Vol. 3, 1977, pp doi:1.1145/ Copyrigh 1 SciRes.
Vehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationChapter 2. Models, Censoring, and Likelihood for Failure-Time Data
Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationIntroduction to Probability and Statistics Slides 4 Chapter 4
Inroducion o Probabiliy and Saisics Slides 4 Chaper 4 Ammar M. Sarhan, asarhan@mahsa.dal.ca Deparmen of Mahemaics and Saisics, Dalhousie Universiy Fall Semeser 8 Dr. Ammar Sarhan Chaper 4 Coninuous Random
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More information556: MATHEMATICAL STATISTICS I
556: MATHEMATICAL STATISTICS I INEQUALITIES 5.1 Concenraion and Tail Probabiliy Inequaliies Lemma (CHEBYCHEV S LEMMA) c > 0, If X is a random variable, hen for non-negaive funcion h, and P X [h(x) c] E
More informationMODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE
Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationAn random variable is a quantity that assumes different values with certain probabilities.
Probabiliy The probabiliy PrA) of an even A is a number in [, ] ha represens how likely A is o occur. The larger he value of PrA), he more likely he even is o occur. PrA) means he even mus occur. PrA)
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationComparison between the Discrete and Continuous Time Models
Comparison beween e Discree and Coninuous Time Models D. Sulsky June 21, 2012 1 Discree o Coninuous Recall e discree ime model Î = AIS Ŝ = S Î. Tese equaions ell us ow e populaion canges from one day o
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
More informationarxiv: v1 [math.ca] 15 Nov 2016
arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationConvergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationSupplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence
Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationStochastic models and their distributions
Sochasic models and heir disribuions Couning cusomers Suppose ha n cusomers arrive a a grocery a imes, say T 1,, T n, each of which akes any real number in he inerval (, ) equally likely The values T 1,,
More informationUnit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationAppendix to Creating Work Breaks From Available Idleness
Appendix o Creaing Work Breaks From Available Idleness Xu Sun and Ward Whi Deparmen of Indusrial Engineering and Operaions Research, Columbia Universiy, New York, NY, 127; {xs2235,ww24}@columbia.edu Sepember
More informationStochastic Model for Cancer Cell Growth through Single Forward Mutation
Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle 31 5-1-2017 Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com
More informationLecture 6: Wiener Process
Lecure 6: Wiener Process Eric Vanden-Eijnden Chapers 6, 7 and 8 offer a (very) brief inroducion o sochasic analysis. These lecures are based in par on a book projec wih Weinan E. A sandard reference for
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationApproximation Algorithms for Unique Games via Orthogonal Separators
Approximaion Algorihms for Unique Games via Orhogonal Separaors Lecure noes by Konsanin Makarychev. Lecure noes are based on he papers [CMM06a, CMM06b, LM4]. Unique Games In hese lecure noes, we define
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationVariational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial
More informationTHE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant).
THE WAVE EQUATION 43. (S) Le u(x, ) be a soluion of he wave equaion u u xx = 0. Show ha Q43(a) (c) is a. Any ranslaion v(x, ) = u(x + x 0, + 0 ) of u(x, ) is also a soluion (where x 0, 0 are consans).
More informationMATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018
MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationThe Arcsine Distribution
The Arcsine Disribuion Chris H. Rycrof Ocober 6, 006 A common heme of he class has been ha he saisics of single walker are ofen very differen from hose of an ensemble of walkers. On he firs homework, we
More informationTransform Techniques. Moment Generating Function
Transform Techniques A convenien way of finding he momens of a random variable is he momen generaing funcion (MGF). Oher ransform echniques are characerisic funcion, z-ransform, and Laplace ransform. Momen
More informationThe Strong Law of Large Numbers
Lecure 9 The Srong Law of Large Numbers Reading: Grimme-Sirzaker 7.2; David Williams Probabiliy wih Maringales 7.2 Furher reading: Grimme-Sirzaker 7.1, 7.3-7.5 Wih he Convergence Theorem (Theorem 54) and
More informationarxiv: v1 [math.fa] 9 Dec 2018
AN INVERSE FUNCTION THEOREM CONVERSE arxiv:1812.03561v1 [mah.fa] 9 Dec 2018 JIMMIE LAWSON Absrac. We esablish he following converse of he well-known inverse funcion heorem. Le g : U V and f : V U be inverse
More informationCash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More informationRepresenting a Signal. Continuous-Time Fourier Methods. Linearity and Superposition. Real and Complex Sinusoids. Jean Baptiste Joseph Fourier
Represening a Signal Coninuous-ime ourier Mehods he convoluion mehod for finding he response of a sysem o an exciaion aes advanage of he lineariy and imeinvariance of he sysem and represens he exciaion
More informationDiebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More informationExponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits
DOI: 0.545/mjis.07.5009 Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis KALPESH S TAILOR Assisan Professor, Deparmen of Saisics, M. K. Bhavnagar Universiy,
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationInventory Control of Perishable Items in a Two-Echelon Supply Chain
Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP. 69-77 69 Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan
More informationHeavy Tails of Discounted Aggregate Claims in the Continuous-time Renewal Model
Heavy Tails of Discouned Aggregae Claims in he Coninuous-ime Renewal Model Qihe Tang Deparmen of Saisics and Acuarial Science The Universiy of Iowa 24 Schae er Hall, Iowa Ciy, IA 52242, USA E-mail: qang@sa.uiowa.edu
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationMost Probable Phase Portraits of Stochastic Differential Equations and Its Numerical Simulation
Mos Probable Phase Porrais of Sochasic Differenial Equaions and Is Numerical Simulaion Bing Yang, Zhu Zeng and Ling Wang 3 School of Mahemaics and Saisics, Huazhong Universiy of Science and Technology,
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationTHE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
More informationSome Ramsey results for the n-cube
Some Ramsey resuls for he n-cube Ron Graham Universiy of California, San Diego Jozsef Solymosi Universiy of Briish Columbia, Vancouver, Canada Absrac In his noe we esablish a Ramsey-ype resul for cerain
More informationEE650R: Reliability Physics of Nanoelectronic Devices Lecture 9:
EE65R: Reliabiliy Physics of anoelecronic Devices Lecure 9: Feaures of Time-Dependen BTI Degradaion Dae: Sep. 9, 6 Classnoe Lufe Siddique Review Animesh Daa 9. Background/Review: BTI is observed when he
More informationLecture 10: The Poincaré Inequality in Euclidean space
Deparmens of Mahemaics Monana Sae Universiy Fall 215 Prof. Kevin Wildrick n inroducion o non-smooh analysis and geomery Lecure 1: The Poincaré Inequaliy in Euclidean space 1. Wha is he Poincaré inequaliy?
More informationNature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.
Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike
More informationThe expectation value of the field operator.
The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationShort Introduction to Fractional Calculus
. Shor Inroducion o Fracional Calculus Mauro Bologna Deparameno de Física, Faculad de Ciencias Universidad de Tarapacá, Arica, Chile email: mbologna@ua.cl Absrac In he pas few years fracional calculus
More informationChristos Papadimitriou & Luca Trevisan November 22, 2016
U.C. Bereley CS170: Algorihms Handou LN-11-22 Chrisos Papadimiriou & Luca Trevisan November 22, 2016 Sreaming algorihms In his lecure and he nex one we sudy memory-efficien algorihms ha process a sream
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationRandom Walk with Anti-Correlated Steps
Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and
More informationApplication of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing
Applicaion of a Sochasic-Fuzzy Approach o Modeling Opimal Discree Time Dynamical Sysems by Using Large Scale Daa Processing AA WALASZE-BABISZEWSA Deparmen of Compuer Engineering Opole Universiy of Technology
More informationRecursive Least-Squares Fixed-Interval Smoother Using Covariance Information based on Innovation Approach in Linear Continuous Stochastic Systems
8 Froniers in Signal Processing, Vol. 1, No. 1, July 217 hps://dx.doi.org/1.2266/fsp.217.112 Recursive Leas-Squares Fixed-Inerval Smooher Using Covariance Informaion based on Innovaion Approach in Linear
More informationThe Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite
American Journal of Operaions Research, 08, 8, 8-9 hp://wwwscirporg/journal/ajor ISSN Online: 60-8849 ISSN Prin: 60-8830 The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process
More informationAvd. Matematisk statistik
Avd Maemaisk saisik TENTAMEN I SF294 SANNOLIKHETSTEORI/EXAM IN SF294 PROBABILITY THE- ORY WEDNESDAY THE 9 h OF JANUARY 23 2 pm 7 pm Examinaor : Timo Koski, el 79 7 34, email: jkoski@khse Tillåna hjälpmedel
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationEE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
More informationThe General Linear Test in the Ridge Regression
ommunicaions for Saisical Applicaions Mehods 2014, Vol. 21, No. 4, 297 307 DOI: hp://dx.doi.org/10.5351/sam.2014.21.4.297 Prin ISSN 2287-7843 / Online ISSN 2383-4757 The General Linear Tes in he Ridge
More informationNotes on Kalman Filtering
Noes on Kalman Filering Brian Borchers and Rick Aser November 7, Inroducion Daa Assimilaion is he problem of merging model predicions wih acual measuremens of a sysem o produce an opimal esimae of he curren
More informationACE 562 Fall Lecture 4: Simple Linear Regression Model: Specification and Estimation. by Professor Scott H. Irwin
ACE 56 Fall 005 Lecure 4: Simple Linear Regression Model: Specificaion and Esimaion by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Simple Regression: Economic and Saisical Model
More informationVanishing Viscosity Method. There are another instructive and perhaps more natural discontinuous solutions of the conservation law
Vanishing Viscosiy Mehod. There are anoher insrucive and perhaps more naural disconinuous soluions of he conservaion law (1 u +(q(u x 0, he so called vanishing viscosiy mehod. This mehod consiss in viewing
More informationExcel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand
Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338
More informationTwo Popular Bayesian Estimators: Particle and Kalman Filters. McGill COMP 765 Sept 14 th, 2017
Two Popular Bayesian Esimaors: Paricle and Kalman Filers McGill COMP 765 Sep 14 h, 2017 1 1 1, dx x Bel x u x P x z P Recall: Bayes Filers,,,,,,, 1 1 1 1 u z u x P u z u x z P Bayes z = observaion u =
More informationInternational Journal of Scientific & Engineering Research, Volume 4, Issue 10, October ISSN
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 900 FUZZY MEAN RESIDUAL LIFE ORDERING OF FUZZY RANDOM VARIABLES J. EARNEST LAZARUS PIRIYAKUMAR 1, A. YAMUNA 2 1.
More informationGeneralized Chebyshev polynomials
Generalized Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, 39 86 Roma, Ialy email: c.cesarano@unineunouniversiy.ne ABSTRACT
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationChapter 4. Location-Scale-Based Parametric Distributions. William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University
Chaper 4 Locaion-Scale-Based Parameric Disribuions William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based on he auhors
More informationHomework sheet Exercises done during the lecture of March 12, 2014
EXERCISE SESSION 2A FOR THE COURSE GÉOMÉTRIE EUCLIDIENNE, NON EUCLIDIENNE ET PROJECTIVE MATTEO TOMMASINI Homework shee 3-4 - Exercises done during he lecure of March 2, 204 Exercise 2 Is i rue ha he parameerized
More informationFractional Method of Characteristics for Fractional Partial Differential Equations
Fracional Mehod of Characerisics for Fracional Parial Differenial Equaions Guo-cheng Wu* Modern Teile Insiue, Donghua Universiy, 188 Yan-an ilu Road, Shanghai 51, PR China Absrac The mehod of characerisics
More informationA new flexible Weibull distribution
Communicaions for Saisical Applicaions and Mehods 2016, Vol. 23, No. 5, 399 409 hp://dx.doi.org/10.5351/csam.2016.23.5.399 Prin ISSN 2287-7843 / Online ISSN 2383-4757 A new flexible Weibull disribuion
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationMA Study Guide #1
MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g()
More information