x 2 x x x x x + x x +2 x
|
|
- Garey Singleton
- 5 years ago
- Views:
Transcription
1 Math 5440: Notes o particle radom walk Aaro Fogelso September 6, 005 Derivatio of the diusio equatio: Imagie that there is a distributio of particles spread alog the x-axis ad that the particles udergo a radom walk. Each particle takes a step x to the right or left every t time uits. We assume that the probability of a particle movig right is p r ad the probability of it movig left is p l ; the steps a particle takes i successive timesteps are idepedet, that is, they have othig to do with earlier steps; ad that the particles move idepedetly of oe aother, so that the motio of oe particle has o eect o ay other particles. x x x x x x + x x + x Figure : Radom walk alog x-axis. Let N(x; t) deote the umber desity, that is, the umber of particles per uit legth. The N(x; t)x is the umber of particles i the iterval (x x; x+ x) at time t. How ca the umber of particles i this iterval chage betwee times t ad t + t? Each particle i the iterval leaves, movig either to the right or to the left. Particles i the iterval to the left may move ito this iterval. So may particles i the iterval to the right. It follows that, N(x; t+t)x = N(x; t)x+p r N(x x; t)x p r N(x; t)x+p l N(x+x; t)x p l N(x; t)x: This equatio ca be rearraged to look like, N(x; t + t) N(x; t) = p r N(x x; t) (p r + p l )N(x; t) + p l N(x + x; t): ()
2 We assume that there are may particles ad that the particle cocetratio N(x; t) varies i a smooth way as x or t chages. Sice x ad t are small, it is reasoable to use Taylor series to approximate N(x; t + t) ad N(x x; t) by expadig about the poit (x; t). N is a fuctio of two idepedet variables (x,t), so, at rst glace, this would seem to require a two-variable Taylor series expasio, but here we have a somewhat easier task, sice for each of the expressios we wish to expad oly oe of the idepedet variables diers from (x; t). We have N(x; t + t) = N(x; t) + N t (x; t)t + N tt(x; t)(t) + :::; N(x+x; t) = N(x; t)+n x (x; t)x+ N xx(x; t)(x) + 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 +::: ad N(x x; t) = N(x; t) N x (x; t)x+ N xx(x; t)(x) 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 +::: Substitutig these expasios ito Eq(), we get N(x; t) + N t (x; t)t + N tt(x; t)(t) + ::: N(x; t) = p r N(x; t) N x (x; t)x + N xx(x; t)(x) 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 + ::: (p r + p l )N(x; t) + p l N(x; t) + N x (x; t)x + N xx(x; t)(x) + 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 + ::: = (p r (p r + p l ) + p l )N(x; t) + ( p r + p l )N x (x; t)x + (p r + p l )N xx (x; t)(x) + 6 (p r p l )N xxx (x; t)(x) (p r + p l )N xxxx (x; t)(x) 4 + ::: We have assumed that each particle takes a step left or right each timestep, so p l + p r =. Let's further assume that the probability of movig left equals the probability of movig right so p l = p r =. The the N x ad N xxx terms above vaish. I the resultig expressio, divide by t ad factor (x) out o the right-had side to get N t + N ttt + ::: = (x) t N xx + N xxxx(x) + ::: :
3 Now imagie that t! 0 ad x! 0, but with (x) t N(x; t) satises kept costat. We d that N t = N xx ; the oe-dimesioal diusio equatio. So for particles doig a radom walk with equal probability of movig left or right, the cocetratio of particles satises the diusio equatio, i the limit that t! 0 ad x! 0, with = (x) t costat. Derivatio of Fick's Law: We have derived the diusio equatio from a macroscopic viewpoit usig coservatio ad Fick's law for the diusive ux, ad from a microscopic viewpoit lookig at particles udergoig a radom walk. Let's relate Fick's law = N x to our microscopic radom walk view. Cosider the ux of particles betwee the two itervals (x x; x + x) ad (x + x; x + 3 x) show i Figure (). x x + x Figure : Flux due to radom walk alog x-axis. Sice there are N(x; t)x particles i the iterval (x x; x+ x), ad each has probabil- ity of movig to the right durig the time step t, the average umber of particles movig to the right is N(x; t)x ad the rate at which particles move to the right is N(x;t)x. t Similarly, with N(x+x; t)x particles i the iterval (x+ x; x+ 3 x), the average rate at which these particles move left is N(x+x;t)x t. We measure ux as positive whe thigs 3
4 move to the right, so the et ux of particles across x + x is = N(x; t)x t = (x) t N(x + x; t)x t N(x + x; t) N(x; t) : x As t! 0 ad x! 0 with = (x) t xed, this equatio implies that = N x ; which is Fick's law. Solutio of the diusio equatio: We ca do more with the radom walk model, icludig obtaiig a solutio to the diusio equatio! Cosider a particle which starts at x = 0 at time t = 0 ad takes a step of size x i each timestep of size t. The particle takes a step to the right with probability p r ad to the left with probability of p l (with p r + p l = ). Let x() deote the locatio of the particle after timesteps. We seek to determie the probability that after steps the particle is at locatio jx. We will deote this by P (j; ) = P rfx() = jxg. Let a be the umber of steps the particle has take to the right, ad b be the umber of steps it has take to the left. The = a+b ad j = a b. For the particle to have laded at jx after steps, it must take a = ( + j)= steps to the right, ad b = ( j)= steps to the left. So let's determie the probability that i steps, the particle takes a steps to the right. This probability is equal to the probability of ay particular path with a steps to the right, times the umber of dieret paths with a steps to the right. The probability of ay particular path with a steps to the right is (p r ) a (p l ) a : The umber of -step paths with exactly a steps to the right is 0 Ca A! = a a!( a)! ; 4
5 that is, ` choose a'. Hece the probability of takig a steps to the right i steps is! a!( a)! (p r) a (p l ) a = B(a; ; p r ) () which is the biomial distributio for trials with probability of `success' (here movig right) p r. I terms of the locatio j, we have P rfx() = jxg = P (j; ) = +j!! j! (p r ) +j (pl ) j : (3) We specialize from ow o to p r = p l = =. Figure(3) shows histograms of P rfx() = jxg as fuctios of j for = 8, 0, ad 00, ad we ca see that they are symmetric as should be expected. (The smooth curves are the Gaussia approximatios derived below.) 0.35 = 8 = 0 = Figure 3: The biomial distributio for = 8, = 0, ad = 00. We ca determie the average displacemet of the particle after steps usig either Eq() or Eq(3). I will use Eq(). If the particle takes a steps to the right i steps, the its locatio is x() = (a ( a))x = (a )x. The average displacemet is therefore < x() >= ( < a > )x; (4) where <> is used to deote the average of the eclosed radom variable. Oly a o the right had side is radom so we compute < a > usig < a >= X k=0 ab(a; ; p r ) = p r = =: (5) 5
6 Hece, usig this i Eq(4) we see that < x() >= 0: The mea squared displacemet is < x() >=< [(a )x] >= (4 < a > 4 < a > + )(x). We kow < a > ad compute < a >= X k=0 a B(a; ; p r ) = (p r ) + p r p l = ; for the case that p l = p r = =. Usig this i the expressio for < x() >, we get < x() >= x : (6) Sice each timestep has duratio t, we ca write our results i terms of time t = t, usig = t=t: < x(t) >= 0; ad where = (x) t i time t is < x(t) >= t t x = x t t = t; as before. The typical distace that a particle moves by the radom walk < x(t) > = = p t: (7) We ca also iterpret this as sayig that the typical time it takes a diusig radom walkig particle to move a distace L is obtaied by solvig L = p t for t, amely t = L. Let's retur to our expressio for the probability of dig the particle at jx after steps as give by Eq(3) ad which I rewrite here for coveiece: P rfx() = jxg = P (j; ) = +j!! j! : We are iterested i the case of may steps >> ad expect that the et umber of steps j away from the startig locatio will be much smaller tha, i.e., j <<. The expressio 6
7 ivolves factorials of large umbers, +j j, ad. There is a very useful approximatio to the factorial of a large umber; it is kow as Stirlig's formula ad is give by l(m!) m + l(m) m + l(): (8) for m >>. We take the atural log of our expressio for P (j; ) to get + j j l fp (j; )g = l(!) l! l! + l( ): We use Stirlig's formula for each of the three factorial terms l(!) + l() + l(); + j l! + j ( + j + ) l + j + l(); ad j l! j ( j + ) l j Substitutig these ito the expressio for lfp (j; )g gives + l(): l fp (j; )g = ( + ) l() + l() (9) ( + j + + j + ) l( ) ( + j + + j + ) l() l ( + ) + j + l j j l + j l j + ( + j + j) l() + l( ): We will make oe further type of approximatio which will allow us to simplify the above expressio. Sice j is presumed to be very small, the followig approximatios are useful: l + j j j + :::; ad l j j j + ::: 7
8 These come from Taylor series expasios of l( + x) for small jxj. Usig these i Eq(9), we d that l fp (j; )g ( + ( + )) l() + ( + ) l() ad therefore it follows that ( + ) l( ) ( + )( j l() l() l( ) = l() + l() l() ; P (j; ) ) j j + l( ) j j e j : (0) So for >> ad j <<, the Biomial distributio is very well approximated by a Gaussia distributio. smooth Gaussia curves i Fig.(3). A idea of how good the approximatio is ca be see from the Istead of cosiderig discrete variables j ad, it is coveiet to express the results i terms of the probability p(x; t)dx of dig the particle i a iterval (x dx=; x + dx=) at time t (here, dx is assumed to be small but a lot bigger tha the tiy radom walk step x). We dee where x = jx, t = t, ad dx x p(x; t)dx P (j; ) dx x ; is the umber of particle ladig sites after steps i the iterval (x dx=; x + dx=). With these chages of variables, we d that p(x; t)dx = t=t or, recallig that (x), we ca write t p(x; t) = 4t = e (x=x) =(t=t) dx x ; () = e x 4t : () ad this is the probability desity for dig a particle which starts at x = 0 at time t = 0 i the iterval (x dx=; x + dx=) at time t. The fuctio p has dimesios of (/legth) ad 8
9 is called a probability desity fuctio (pdf). It has the iterpretatio that the probability of dig the particle i the iterval (x ; x ) at time t is give by the itegral Z x x p(x; t)dx: Usig the fact that Z e s ds = p you ca verify that R p(x; t)dx = for ay t. We have see the fuctio p(x; t) give i Eq() before. It is the fuctio that I called the fudametal solutio of the diusio equatio; we have derived it by lookig at particles udergoig radom walk alog the x-axis from a startig locatio at x = 0! 9
MATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationThe Poisson Process *
OpeStax-CNX module: m11255 1 The Poisso Process * Do Johso This work is produced by OpeStax-CNX ad licesed uder the Creative Commos Attributio Licese 1.0 Some sigals have o waveform. Cosider the measuremet
More informationHOMEWORK #10 SOLUTIONS
Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationMath 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 informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationx = Pr ( X (n) βx ) =
Exercise 93 / page 45 The desity of a variable X i i 1 is fx α α a For α kow let say equal to α α > fx α α x α Pr X i x < x < Usig a Pivotal Quatity: x α 1 < x < α > x α 1 ad We solve i a similar way as
More informationDiscrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15
CS 70 Discrete Mathematics ad Probability Theory Sprig 2012 Alistair Siclair Note 15 Some Importat Distributios The first importat distributio we leared about i the last Lecture Note is the biomial distributio
More information1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
More informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More information18.01 Calculus Jason Starr Fall 2005
Lecture 18. October 5, 005 Homework. Problem Set 5 Part I: (c). Practice Problems. Course Reader: 3G 1, 3G, 3G 4, 3G 5. 1. Approximatig Riema itegrals. Ofte, there is o simpler expressio for the atiderivative
More informationSTAT 516 Answers Homework 6 April 2, 2008 Solutions by Mark Daniel Ward PROBLEMS
STAT 56 Aswers Homework 6 April 2, 28 Solutios by Mark Daiel Ward PROBLEMS Chapter 6 Problems 2a. The mass p(, correspods to either o the irst two balls beig white, so p(, 8 7 4/39. The mass p(, correspods
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationAreas and Distances. We can easily find areas of certain geometric figures using well-known formulas:
Areas ad Distaces We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate the area of the regio
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationExpectation 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 informationLecture 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 information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationDiscrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions
CS 70 Discrete Mathematics for CS Sprig 2005 Clacy/Wager Notes 21 Some Importat Distributios Questio: A biased coi with Heads probability p is tossed repeatedly util the first Head appears. What is the
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)
Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationKurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version)
Kurskod: TAMS Provkod: TENB 2 March 205, 4:00-8:00 Examier: Xiagfeg Yag (Tel: 070 2234765). Please aswer i ENGLISH if you ca. a. You are allowed to use: a calculator; formel -och tabellsamlig i matematisk
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationMath 257: Finite difference methods
Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationFor example suppose we divide the interval [0,2] into 5 equal subintervals of length
Math 1206 Calculus Sec 1: Estimatig with Fiite Sums Abbreviatios: wrt with respect to! for all! there exists! therefore Def defiitio Th m Theorem sol solutio! perpedicular iff or! if ad oly if pt poit
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationExercises and Problems
HW Chapter 4: Oe-Dimesioal Quatum Mechaics Coceptual Questios 4.. Five. 4.4.. is idepedet of. a b c mu ( E). a b m( ev 5 ev) c m(6 ev ev) Exercises ad Problems 4.. Model: Model the electro as a particle
More informationA widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α
Nice plottig of proteis: I A widely used display of protei shapes is based o the coordiates of the alpha carbos - - C α -s. The coordiates of the C α -s are coected by a cotiuous curve that roughly follows
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More informationChapter 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 informationLecture 3: Catalan Numbers
CCS Discrete Math I Professor: Padraic Bartlett Lecture 3: Catala Numbers Week 3 UCSB 2014 I this week, we start studyig specific examples of commoly-occurrig sequeces of umbers (as opposed to the more
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationThe Poisson Distribution
MATH 382 The Poisso Distributio Dr. Neal, WKU Oe of the importat distributios i probabilistic modelig is the Poisso Process X t that couts the umber of occurreces over a period of t uits of time. This
More informationFluid Physics 8.292J/12.330J % (1)
Fluid Physics 89J/133J Problem Set 5 Solutios 1 Cosider the flow of a Euler fluid i the x directio give by for y > d U = U y 1 d for y d U + y 1 d for y < This flow does ot vary i x or i z Determie the
More informationTopic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.
Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationExponential 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 informationEstimation 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 informationP1 Chapter 8 :: Binomial Expansion
P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationMath 116 Second Exam
Math 6 Secod Exam November, 6 Name: Exam Solutios Istructor: Sectio:. Do ot ope this exam util you are told to do so.. This exam has 9 pages icludig this cover. There are 8 problems. Note that the problems
More informationThe Binomial Theorem
The Biomial Theorem Robert Marti Itroductio The Biomial Theorem is used to expad biomials, that is, brackets cosistig of two distict terms The formula for the Biomial Theorem is as follows: (a + b ( k
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationMath 21C Brian Osserman Practice Exam 2
Math 1C Bria Osserma Practice Exam 1 (15 pts.) Determie the radius ad iterval of covergece of the power series (x ) +1. First we use the root test to determie for which values of x the series coverges
More informationGeneralized Semi- Markov Processes (GSMP)
Geeralized Semi- Markov Processes (GSMP) Summary Some Defiitios Markov ad Semi-Markov Processes The Poisso Process Properties of the Poisso Process Iterarrival times Memoryless property ad the residual
More informationMonte 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 informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationJEE ADVANCED 2013 PAPER 1 MATHEMATICS
Oly Oe Optio Correct Type JEE ADVANCED 0 PAPER MATHEMATICS This sectio cotais TEN questios. Each has FOUR optios (A), (B), (C) ad (D) out of which ONLY ONE is correct.. The value of (A) 5 (C) 4 cot cot
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationCHAPTER 8 SYSTEMS OF PARTICLES
CHAPTER 8 SYSTES OF PARTICLES CHAPTER 8 COLLISIONS 45 8. CENTER OF ASS The ceter of mass of a system of particles or a rigid body is the poit at which all of the mass are cosidered to be cocetrated there
More informationPower Series Expansions of Binomials
Power Series Expasios of Biomials S F Ellermeyer April 0, 008 We are familiar with expadig biomials such as the followig: ( + x) = + x + x ( + x) = + x + x + x ( + x) 4 = + 4x + 6x + 4x + x 4 ( + x) 5
More informationSECTION 2 Electrostatics
SECTION Electrostatics This sectio, based o Chapter of Griffiths, covers effects of electric fields ad forces i static (timeidepedet) situatios. The topics are: Electric field Gauss s Law Electric potetial
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationDiscrete 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 informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationMath 116 Second Midterm November 13, 2017
Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.
More informationMath 105: Review for Final Exam, Part II - SOLUTIONS
Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio f(x) = x 3 lx o the iterval [/e, e ]. (a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum
More informationCOMM 602: Digital Signal Processing
COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationIIT JAM Mathematical Statistics (MS) 2006 SECTION A
IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationPractice Problems: Taylor and Maclaurin Series
Practice Problems: Taylor ad Maclauri Series Aswers. a) Start by takig derivatives util a patter develops that lets you to write a geeral formula for the -th derivative. Do t simplify as you go, because
More informationSOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker
SOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker CHAPTER 9. POINT ESTIMATION 9. Covergece i Probability. The bases of poit estimatio have already bee laid out i previous chapters. I chapter 5
More informationLecture 2: Poisson Sta*s*cs Probability Density Func*ons Expecta*on and Variance Es*mators
Lecture 2: Poisso Sta*s*cs Probability Desity Fuc*os Expecta*o ad Variace Es*mators Biomial Distribu*o: P (k successes i attempts) =! k!( k)! p k s( p s ) k prob of each success Poisso Distributio Note
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationPhysics 116A Solutions to Homework Set #9 Winter 2012
Physics 116A Solutios to Homework Set #9 Witer 1 1. Boas, problem 11.3 5. Simplify Γ( 1 )Γ(4)/Γ( 9 ). Usig xγ(x) Γ(x + 1) repeatedly, oe obtais Γ( 9) 7 Γ( 7) 7 5 Γ( 5 ), etc. util fially obtaiig Γ( 9)
More informationLecture 19. sup y 1,..., yn B d n
STAT 06A: Polyomials of adom Variables Lecture date: Nov Lecture 19 Grothedieck s Iequality Scribe: Be Hough The scribes are based o a guest lecture by ya O Doell. I this lecture we prove Grothedieck s
More informationLecture 18: Sampling distributions
Lecture 18: Samplig distributios I may applicatios, the populatio is oe or several ormal distributios (or approximately). We ow study properties of some importat statistics based o a radom sample from
More informationPHY4905: Nearly-Free Electron Model (NFE)
PHY4905: Nearly-Free Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. No-degeerate eigestates
More informationExercise 4.3 Use the Continuity Theorem to prove the Cramér-Wold Theorem, Theorem. (1) φ a X(1).
Assigmet 7 Exercise 4.3 Use the Cotiuity Theorem to prove the Cramér-Wold Theorem, Theorem 4.12. Hit: a X d a X implies that φ a X (1) φ a X(1). Sketch of solutio: As we poited out i class, the oly tricky
More informationTopic 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 informationLinear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationZ ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew
Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe
More informationPhysics 232 Gauge invariance of the magnetic susceptibilty
Physics 232 Gauge ivariace of the magetic susceptibilty Peter Youg (Dated: Jauary 16, 2006) I. INTRODUCTION We have see i class that the followig additioal terms appear i the Hamiltoia o addig a magetic
More informationMATH Exam 1 Solutions February 24, 2016
MATH 7.57 Exam Solutios February, 6. Evaluate (A) l(6) (B) l(7) (C) l(8) (D) l(9) (E) l() 6x x 3 + dx. Solutio: D We perform a substitutio. Let u = x 3 +, so du = 3x dx. Therefore, 6x u() x 3 + dx = [
More informationMath 116 Final Exam December 12, 2014
Math 6 Fial Exam December 2, 24 Name: EXAM SOLUTIONS Istructor: Sectio:. Do ot ope this exam util you are told to do so. 2. This exam has 4 pages icludig this cover. There are 2 problems. Note that the
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More information