Introduction to Probability and Statistics Slides 4 Chapter 4
|
|
- Brooke Sherman
- 6 years ago
- Views:
Transcription
1 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
2 Dr. Ammar Sarhan Chaper 4 Coninuous Random Variables and Probabiliy Disribuions
3 4. Coninuous Random Variables and Probabiliy Disribuions Coninuous Random Variables A random variable is coninuous if () is se of possible values is an enire inerval of numbers; () P( c) for any number c. Example: () The deph of a chosen locaion; () The lifeime of a produc; () The waiing ime spen by a cusomer o receive his/her serves.
4 Probabiliy Disribuion Dr. Ammar Sarhan 4 Le be a coninuous rv. Then a probabiliy disribuion or probabiliy densiy funcion (pdf) of is a funcion f (x) such ha for any wo numbers a and b, P ( a b) f ( x) dx This gives he probabiliy ha akes on a value in he inerval [a, b]. I also gives he area under he densiy curve. b a P(a b) f (x) a b x
5 Probabiliy Densiy Funcion Dr. Ammar Sarhan 5 The pdf f (x) saisfies he following condiions:. f (x) for all values of x.. The area beween he graph of f (x) and he x-axis is equal o. P( ) f ( x) dx If is a coninuous rv hen For any even A defined on, P ( A) f ( x) dx x A P(a b) P(a < b) P(a < b) P(a < < b)
6 Dr. Ammar Sarhan 6 Example: Le be a rv wih he following pdf f ( x) 6 x < 6 oherwise 6 f (x) P( 6) dx 6 P(9 8) Because whenever a b 6, P(a b) depends only on he widh b-a of he inerval, is said o have a uniform disribuion.
7 Dr. Ammar Sarhan 7 Uniform Disribuion A coninuous rv is said o have a uniform disribuion on he inerval [A, B] if he pdf of is Example: (Prob. 5, p. 5) is he ime elapses beween he end of he hour and he end of he lecure. a) Find k. Since, k f ( x) dx 8 k x x < f ( x) oherwise k k x dx k x
8 b) Wha is he probabiliy ha he lecure ends wihin min of he end of he hour? Dr. Ammar Sarhan 8 k P( ) f ( x) dx k x dx k x /8 8.5 c) Wha is he probabiliy ha he lecure coninuous beyond he hour for beween 6 and 9 sec? min P(.5) 8.5 f ( x) dx.5 [( ) ] c) Wha is he probabiliy ha he lecure coninuous for a leas 9 sec beyond he end of he hour? min P(.5 ).5 8 f ( x) dx.5 k x k x dx [ ] dx k x k x.5.5
9 4. Cumulaive disribuion funcions and Expeced values Dr. Ammar Sarhan 9 The Cumulaive Disribuion Funcion The cumulaive disribuion funcion, F (x) for a coninuous rv is defined for every number x by F( x) x ( x) f ( y dy P ) For each x, F(x) is he area under he densiy curve o he lef of x. Example: (4.6, p. 7)
10 Using F(x) o Compue Probabiliies Dr. Ammar Sarhan Le be a coninuous rv wih pdf f(x) and cdf F(x). Then for any number a, P( > a) - F(a) and for any numbers a and b wih a < b, Example: (4.7, p. 8) P(a b) F(b) - F(a)
11 Obaining f(x) from F(x) Dr. Ammar Sarhan If is a coninuous rv wih pdf f(x) and cdf F(x), hen a every number x for which he derivaive F (x) exiss F (x) f (x). Example: (4.8, p. 9) Perceniles When we say ha an individual s es score was a he 85 h percenile of he populaion, we mean ha 85% of all populaion scores were below ha score and 5% were above. Le p be a number beween and. The (p)h percenile of he disribuion of a coninuous rv denoed by η(p), is defined by p P η ( p) ( η( p) ) F( η( p)) f ( y) dy
12 Dr. Ammar Sarhan Shaded area p F (x) p F(η(p)) η(p) x η(p) x Example: Le be a rv wih he following pdf x x f ( x) oherwise Find he (p)% percenile of? Firs, he cdf F x x x ( x) f ( y) dy y dy y x for x
13 The (p)h percenile is he soluion of he following eqn: Dr. Ammar Sarhan p F( η( p)) Then, he (p)h percenile is p [ η( p) ] η For he 5h percenile, p.5, η.5.77 p f (x) F (x).5 F(η(.5)) η(.5).77 x η(.5).77 x
14 Dr. Ammar Sarhan 4 Median The median of a coninuous disribuion, denoed by, is he 5h percenile. So ~ μ saisfies.5 F( ~ μ ). Tha is, half he area under he densiy curve is o he lef of ~ μ. Noice: For he disribuion wih symmeric pdf he median equals he poin of symmery. μ ~ f(x) f(x) f(x) μ ~ μ ~ Medians of symmeric disribuions μ ~
15 Expeced Value Dr. Ammar Sarhan 5 The expeced or mean value of a coninuous rv wih pdf f (x) is μ E( ) x f ( x) dx Example: In he previous example, find E(). E ( ) x f ( x) dx x dx y Expeced Value of h() If is a coninuous rv wih pdf f(x) and h(x) is any funcion of, hen μ [ h( )] E h( x) f ( x dx h ( ) )
16 Variance and Sandard Deviaion Dr. Ammar Sarhan 6 The variance of coninuous rv wih pdf f(x) and mean μ is V ( ) σ E[( μ) ] ( x μ) f ( x) dx. The sandard deviaion (SD) of is σ σ Shor-cu Formula for Variance V ( ) ( ) E μ
17 Example: In he previous example, find ) V(), ) E[ + ] ) V[ + ] Dr. Ammar Sarhan 7 ) V ( ) ( ) E μ We have μ E( ) and Then E 4 ( ) x f ( x) dx x dx 4 y 4 V ( ) ) E[ + ] E() + (/) + 4 ) V[ + ] V() 9 V() 9 (5/8)
18 Dr. Ammar Sarhan 8 Exponenial Disribuion A coninuous rv has an exponenial disribuion wih parameer λ> if he pdf is λ e f ( x; λ) λ x x oherwise Mean and Variance Le be a rv having he exponenial disribuion wih parameer λ. Then E( ) V ( ) λ λ Cdf of exponenial Disribuion F( x; λ) e λ x x oherwise
19 Dr. Ammar Sarhan The Memoryless propery of exponenial Disribuion Le has an exponenial disribuion wih parameer λ>. Then ) ( ) ( P P + ( ) ( ) ( ) ( ) ) ( P P P + + ( ) ( ) ( ) ( ) F F P P + + ( ) ( ) ( ) ( ) e e e e e λ λ λ λ λ + + ) ( ) ( P F Proof: 9
20 Dr. Ammar Sarhan Example: Suppose ha he response ime a a cerain on-line compuer erminal (he elapsed ime beween he end of a user s inquiry and he beginning of he sysem s response o ha inquiry) has an exponenial disribuion wih expeced response ime equal o 5 esc. Find ) Probabiliy ha he response ime is a mos sec? Since E() 5 /λ, hen λ /5. P( ) F() - e -.() - e ) Probabiliy ha he response ime is beween 5 and sec? P(5 ) F() F(5) e -.(5) -e -.() e - -e -. ) Assume ha one user is waied for sec, wha is he probabiliy ha he will ge he sysem s response wihin he nex 5sec?
21 Dr. Ammar Sarhan Applicaions of he Exponenial Disribuion The exponenial disribuion is frequenly used as a model for he disribuion of he occurrence of successive evens, such as ) Cusomers arrivals a a service faciliy, ) Calls coming o a swichboard. The reason for his is ha he exponenial disribuion is closely relaed o he Poisson process. Proposiion Suppose ha he number of evens occurring in any ime inerval of lengh has a Poisson disribuion wih parameer α (where α is he expeced number of evens occurring in uni of ime) and ha he numbers of occurrences in nonoverlapping inervals are independen of one anoher. Then he disribuion of elapsed ime beween he occurrences of wo successive evens is exponenial wih parameer λ α.
22 Example: Dr. Ammar Sarhan Suppose ha calls are received a 4-hour holine according o a Poisson process wih rae α.5 per day. Compue. The probabiliy ha more han days elapse beween wo successive calls (wo calls)?. The expeced ime beween wo successive calls.. Given ha hey have jus received a call, wha is he probabiliy ha hey will receive a call wihin he nex 6 hours? Soluion: Le denoe he number of days beween successive calls. Then ~ Exp(.5). Using his informaion, one can easily answer he above hree quesions.
23 4. The Normal Disribuion Dr. Ammar Sarhan A coninuous rv is said o have a normal disribuion wih parameers μ and σ, where - < μ < and < σ, if he pdf of is f ( x; μ, σ ) e σ π ( x μ ) /(σ ), < x < Sandard Normal Disribuions The normal disribuion wih parameer values μ and σ is called a sandard normal disribuion. The random variable is denoed by Z. The pdf is f ( x;,) e π x /, < x <
24 Dr. Ammar Sarhan 4 The cdf is z Φ ( z) P( Z z) f ( y;,) dy Sandard Normal Cumulaive Areas
25 Dr. Ammar Sarhan 5 Sandard Normal Disribuion Le Z be he sandard normal variable. Find (from able) a. P(Z <.85) Area o he lef of b. P(Z >.) - P(Z.).94 c. P(-. Z.78) P(Z.78) P(Z -.)
26 z α Noaion Dr. Ammar Sarhan 6 z α will denoe he value on he measuremen axis for which he area under he z curve lies o he righ of z α is α. z curve P(Z <-z α ) α z α Since α of he area under he z curve lies o he righ of z α, hen -α of he area lies o is lef. Thus, z α is he (-α)h percenile of he sandard normal disribuion. By symmery he area under SNC o he lef of z α is also α. The z α s are usually referred o as z criical values. Table 4. liss he mos useful perceniles and values z α.
27 Example: The z.5 is he (-.5)h percenile, so z The area under he SND curve o he lef of z.5 is also.5. Dr. Ammar Sarhan 7 z curve z z h percenile
28 Dr. Ammar Sarhan 8 Example: Le Z be he sandard normal variable. Find z if a. P(Z < z).978 Look a he able and find an enry.978 hen read back o find z.46 b. P( z < Z < z).8 P(-z < Z < z ) P(Z<z) P(Z<-z) P(Z<z) P(Z>z) P(Z<z) [-P(Z<z)] P(Z<z) +P(Z<z) P(Z < z).8+ P(Z < z) P(Z < z).8 /.966 z.
29 Nonsandard Normal Disribuions Dr. Ammar Sarhan 9 If has a normal disribuion wih mean μ and sandard deviaions σ, shorly N(μ, σ ), hen has a sandard normal disribuion. Example: Le be a normal random variable wih μ 8 and σ. Find P( 65)?
30 Example: A paricular rash shown up a an elemenary school. I has been deermined ha he lengh of ime ha he rash will las is normally disribued wih μ 6 days and σ.5 days. Find he probabiliy ha for a suden seleced a random, he rash will las for beween.75 and 9 days? Dr. Ammar Sarhan Suppose ha ~ N(6,.5), find P(.75 9)?
31 Perceniles of an Arbirary Normal Disribuion Dr. Ammar Sarhan (p)h percenile for normal(μ, σ ) (p)h percenile μ + σ sandard normal Example: The amoun of disilled waer dispensed by a cerain machine is normally disribued wih mean value 64 oz and sandard deviaion.78 oz. Wha conainer size c will ensure ha overflow occurs only.5% of he ime? Le denoe he amoun dispensed, hen ~ N(64,.78). The desired condiion is ha P( > c).5, or, equivalenly ha P( c).995. Thus, c is he 99 h percenile of N(64,.78). Since 99 h of Z is z..58, hen c 64 + (.58) (.78) oz
32 Normal Approximaion o he Binomial Disribuion Dr. Ammar Sarhan Le be a binomial rv based on n rials, each wih probabiliy of success p. If he binomial probabiliy hisogram is no oo skewed, may be approximaed by a normal disribuion wih μ np and σ npq. P( x) x +.5 np Φ npq In pracice, his approximaion is adequae provided ha boh np and nq. Example: A a paricular small college he pass rae of Inermediae Algebra is 7%. If 5 sudens enrol in a semeser deermine he probabiliy ha a mos 75 sudens pass.
33 Dr. Ammar Sarhan Le denoe he number of sudens pass he es. Then, ~ Bino(5,.7) Soluion and he desired probabiliy is P( 75 ) μ np 5 (.7) 6, and n(-p) 5 (.8) 4, σ npq 5(.7)(.8) Then, approximaely ~ N(6, ) P( 75) Φ Φ (.55).994
Stochastic 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 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 informationVehicle 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 informationBasic notions of probability theory (Part 2)
Basic noions of probabiliy heory (Par 2) Conens o Basic Definiions o Boolean Logic o Definiions of probabiliy o Probabiliy laws o Random variables o Probabiliy Disribuions Random variables Random variables
More informationRandom variables. A random variable X is a function that assigns a real number, X(ζ), to each outcome ζ in the sample space of a random experiment.
Random variables Some random eperimens may yield a sample space whose elemens evens are numbers, bu some do no or mahemaical purposes, i is desirable o have numbers associaed wih he oucomes A random variable
More informationin Engineering Prof. Dr. Michael Havbro Faber ETH Zurich, Switzerland Swiss Federal Institute of Technology
Risk and Saey in Engineering Pro. Dr. Michael Havbro Faber ETH Zurich, Swizerland Conens o Today's Lecure Inroducion o ime varian reliabiliy analysis The Poisson process The ormal process Assessmen o he
More informationReliability of Technical Systems
eliabiliy of Technical Sysems Main Topics Inroducion, Key erms, framing he problem eliabiliy parameers: Failure ae, Failure Probabiliy, Availabiliy, ec. Some imporan reliabiliy disribuions Componen reliabiliy
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
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 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 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 informationECE 510 Lecture 4 Reliability Plotting T&T Scott Johnson Glenn Shirley
ECE 5 Lecure 4 Reliabiliy Ploing T&T 6.-6 Sco Johnson Glenn Shirley Funcional Forms 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley Reliabiliy Funcional Forms Daa Model (funcional form) Choose funcional form for
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
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 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 informationECE 510 Lecture 4 Reliability Plotting T&T Scott Johnson Glenn Shirley
ECE 5 Lecure 4 Reliabiliy Ploing T&T 6.-6 Sco Johnson Glenn Shirley Funcional Forms 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 2 Reliabiliy Funcional Forms Daa Model (funcional form) Choose funcional form
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
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 informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
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 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 informationPROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES
PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is
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 informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 8 4 6 Diameer The able above gives he measuremens of he B(x)
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 informationStationary Distribution. Design and Analysis of Algorithms Andrei Bulatov
Saionary Disribuion Design and Analysis of Algorihms Andrei Bulaov Algorihms Markov Chains 34-2 Classificaion of Saes k By P we denoe he (i,j)-enry of i, j Sae is accessible from sae if 0 for some k 0
More informationDiscrete Markov Processes. 1. Introduction
Discree Markov Processes 1. Inroducion 1. Probabiliy Spaces and Random Variables Sample space. A model for evens: is a family of subses of such ha c (1) if A, hen A, (2) if A 1, A 2,..., hen A1 A 2...,
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 informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationMaintenance Models. Prof. Robert C. Leachman IEOR 130, Methods of Manufacturing Improvement Spring, 2011
Mainenance Models Prof Rober C Leachman IEOR 3, Mehods of Manufacuring Improvemen Spring, Inroducion The mainenance of complex equipmen ofen accouns for a large porion of he coss associaed wih ha equipmen
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3
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 informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
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 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 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 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 informationSTRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN
Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN 001-004 OBARA, Takashi * Absrac The
More informationT L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
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 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 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 informationLecture 33: November 29
36-705: Inermediae Saisics Fall 2017 Lecurer: Siva Balakrishnan Lecure 33: November 29 Today we will coninue discussing he boosrap, and hen ry o undersand why i works in a simple case. In he las lecure
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 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 information5.2. The Natural Logarithm. Solution
5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,
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 informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationBasic definitions and relations
Basic definiions and relaions Lecurer: Dmiri A. Molchanov E-mail: molchan@cs.u.fi hp://www.cs.u.fi/kurssi/tlt-2716/ Kendall s noaion for queuing sysems: Arrival processes; Service ime disribuions; Examples.
More informationRandom Processes 1/24
Random Processes 1/24 Random Process Oher Names : Random Signal Sochasic Process A Random Process is an exension of he concep of a Random variable (RV) Simples View : A Random Process is a RV ha is a Funcion
More informationTMA 4265 Stochastic Processes
TMA 4265 Sochasic Processes Norges eknisk-naurvienskapelige universie Insiu for maemaiske fag Soluion - Exercise 8 Exercises from he ex book 5.2 The expeced service ime for one cusomer is 1/µ, due o he
More informationACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H.
ACE 564 Spring 2006 Lecure 7 Exensions of The Muliple Regression Model: Dumm Independen Variables b Professor Sco H. Irwin Readings: Griffihs, Hill and Judge. "Dumm Variables and Varing Coefficien Models
More information72 Calculus and Structures
72 Calculus and Srucures CHAPTER 5 DISTANCE AND ACCUMULATED CHANGE Calculus and Srucures 73 Copyrigh Chaper 5 DISTANCE AND ACCUMULATED CHANGE 5. DISTANCE a. Consan velociy Le s ake anoher look a Mary s
More informationExam 3 Review (Sections Covered: , )
19 Exam Review (Secions Covered: 776 8184) 1 Adieisloadedandihasbeendeerminedhaheprobabiliydisribuionassociaedwih he experimen of rolling he die and observing which number falls uppermos is given by he
More informationMath 115 Final Exam December 14, 2017
On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):
More informationInnova Junior College H2 Mathematics JC2 Preliminary Examinations Paper 2 Solutions 0 (*)
Soluion 3 x 4x3 x 3 x 0 4x3 x 4x3 x 4x3 x 4x3 x x 3x 3 4x3 x Innova Junior College H Mahemaics JC Preliminary Examinaions Paper Soluions 3x 3 4x 3x 0 4x 3 4x 3 0 (*) 0 0 + + + - 3 3 4 3 3 3 3 Hence x or
More informationON THE NUMBER OF FAMILIES OF BRANCHING PROCESSES WITH IMMIGRATION WITH FAMILY SIZES WITHIN RANDOM INTERVAL
ON THE NUMBER OF FAMILIES OF BRANCHING PROCESSES ITH IMMIGRATION ITH FAMILY SIZES ITHIN RANDOM INTERVAL Husna Hasan School of Mahemaical Sciences Universii Sains Malaysia, 8 Minden, Pulau Pinang, Malaysia
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
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 informationComparing Means: t-tests for One Sample & Two Related Samples
Comparing Means: -Tess for One Sample & Two Relaed Samples Using he z-tes: Assumpions -Tess for One Sample & Two Relaed Samples The z-es (of a sample mean agains a populaion mean) is based on he assumpion
More informationRight tail. Survival function
Densiy fi (con.) Lecure 4 The aim of his lecure is o improve our abiliy of densiy fi and knowledge of relaed opics. Main issues relaed o his lecure are: logarihmic plos, survival funcion, HS-fi mixures,
More information) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha
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 informationACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H.
ACE 56 Fall 005 Lecure 5: he Simple Linear Regression Model: Sampling Properies of he Leas Squares Esimaors by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Inference in he Simple
More information04. Kinetics of a second order reaction
4. Kineics of a second order reacion Imporan conceps Reacion rae, reacion exen, reacion rae equaion, order of a reacion, firs-order reacions, second-order reacions, differenial and inegraed rae laws, Arrhenius
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 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 informationMath Final Exam Solutions
Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,
More informationAsymptotic Equipartition Property - Seminar 3, part 1
Asympoic Equipariion Propery - Seminar 3, par 1 Ocober 22, 2013 Problem 1 (Calculaion of ypical se) To clarify he noion of a ypical se A (n) ε and he smalles se of high probabiliy B (n), we will calculae
More informationHomework 4 (Stats 620, Winter 2017) Due Tuesday Feb 14, in class Questions are derived from problems in Stochastic Processes by S. Ross.
Homework 4 (Sas 62, Winer 217) Due Tuesday Feb 14, in class Quesions are derived from problems in Sochasic Processes by S. Ross. 1. Le A() and Y () denoe respecively he age and excess a. Find: (a) P{Y
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 informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More informationLearning a Class from Examples. Training set X. Class C 1. Class C of a family car. Output: Input representation: x 1 : price, x 2 : engine power
Alpaydin Chaper, Michell Chaper 7 Alpaydin slides are in urquoise. Ehem Alpaydin, copyrigh: The MIT Press, 010. alpaydin@boun.edu.r hp://www.cmpe.boun.edu.r/ ehem/imle All oher slides are based on Michell.
More informationCHERNOFF DISTANCE AND AFFINITY FOR TRUNCATED DISTRIBUTIONS *
haper 5 HERNOFF DISTANE AND AFFINITY FOR TRUNATED DISTRIBUTIONS * 5. Inroducion In he case of disribuions ha saisfy he regulariy condiions, he ramer- Rao inequaliy holds and he maximum likelihood esimaor
More information1998 Calculus AB Scoring Guidelines
AB{ / BC{ 1999. The rae a which waer ows ou of a pipe, in gallons per hour, is given by a diereniable funcion R of ime. The able above shows he rae as measured every hours for a {hour period. (a) Use a
More informationStatistical Distributions
Saisical Disribuions 1 Discree Disribuions 1 The uniform disribuion A random variable (rv) X has a uniform disribuion on he n-elemen se A = {x 1,x 2,,x n } if P (X = x) =1/n whenever x is in he se A The
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 informationEstimation of Poses with Particle Filters
Esimaion of Poses wih Paricle Filers Dr.-Ing. Bernd Ludwig Chair for Arificial Inelligence Deparmen of Compuer Science Friedrich-Alexander-Universiä Erlangen-Nürnberg 12/05/2008 Dr.-Ing. Bernd Ludwig (FAU
More informationA Bayesian Approach to Spectral Analysis
Chirped Signals A Bayesian Approach o Specral Analysis Chirped signals are oscillaing signals wih ime variable frequencies, usually wih a linear variaion of frequency wih ime. E.g. f() = A cos(ω + α 2
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationSensors, Signals and Noise
Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1 Noise Descripion Noise Waveforms and Samples Saisics of Noise
More informationWednesday, November 7 Handout: Heteroskedasticity
Amhers College Deparmen of Economics Economics 360 Fall 202 Wednesday, November 7 Handou: Heeroskedasiciy Preview Review o Regression Model o Sandard Ordinary Leas Squares (OLS) Premises o Esimaion Procedures
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 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 informationAnswers to Exercises in Chapter 7 - Correlation Functions
M J Robers - //8 Answers o Exercises in Chaper 7 - Correlaion Funcions 7- (from Papoulis and Pillai) The random variable C is uniform in he inerval (,T ) Find R, ()= u( C), ()= C (Use R (, )= R,, < or
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 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 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 informationChapter 14 Wiener Processes and Itô s Lemma. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull
Chaper 14 Wiener Processes and Iô s Lemma Copyrigh John C. Hull 014 1 Sochasic Processes! Describes he way in which a variable such as a sock price, exchange rae or ineres rae changes hrough ime! Incorporaes
More informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
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 informationChapter 2 Basic Reliability Mathematics
Chaper Basic Reliabiliy Mahemaics The basics of mahemaical heory ha are relevan o he sudy of reliabiliy and safey engineering are discussed in his chaper. The basic conceps of se heory and probabiliy heory
More informationConcept of Random Variables
Concep of andom Variables DOMAIN S is he enire ais ANE S is he posiive ais. Hence iven wo ses of numbers S and S S hen o every we assign a number () belonging o S S The generalizaion ( ) iven wo ses of
More informationNavneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi
Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec
More informationChapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationState-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter
Sae-Space Models Iniializaion, Esimaion and Smoohing of he Kalman Filer Iniializaion of he Kalman Filer The Kalman filer shows how o updae pas predicors and he corresponding predicion error variances when
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 informationContinuous Time Markov Chain (Markov Process)
Coninuous Time Markov Chain (Markov Process) The sae sace is a se of all non-negaive inegers The sysem can change is sae a any ime ( ) denoes he sae of he sysem a ime The random rocess ( ) forms a coninuous-ime
More information