4.1. Probability Density Functions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "4.1. Probability Density Functions"

Transcription

1 STT Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of continuous rndom vriles. Description of continuous rndom vriles (crv) A continuous rndom vrile tkes on n uncountly infinite numer of vlues. The set of possile vlues of every crv is n entire intervl, finite or infinite. Continuous rndom vriles usully re mesurements. Exmples of continuous rndom vriles Weight nd height of people. Depth mesure in certin loction of lke. The mount of sugr in n ornge. The time required to run mile. The computer time (in seconds) required to process certin progrm. Mesurement of ny physicl quntity with dditive rndom error (noise). The mount of rin flls in the certin city over fixed period of time. Definition Let X e continuous rndom vrile. The proility density function of X is function f such tht for ny two numers nd with P X f x dx 1

2 STT Geometric interprettion of P X f x dx, P X Are under the curve y f x ove the intervl Properties of Proility Density Function. f x f xdx 1 for ll x Exmple (#1 p.14 textook) The current in certin circuit s mesured y n mmeter is continuous rndom vrile X with the following density function: f x.7x x otherwise ) Verify tht the totl re under the density curve is indeed 1. ) Clculte P X 4. 4 P X 4.7x dx.6 c) Clculte P. X P. X 4..7x dx. d) Clculte P X P X 4..7x dx

3 STT Continuous rndom vrile X tkes every its vlue with proility. P X x ues x for ll possile vl ecuse c c P X c f x dx lim f x dx c c Exmple (continution of 4.1.6) P X 4 nd 4 P. X 4. P X 4. nd P X 4.. P X ; nd P. X 4. In ech pir ove, the numers re equl Uniform Distriution. ; Definition. A continuous rv X is sid to hve uniform distriution on the intervl, if the pdf of X is 1 x f x;, otherwise Exmple Assume the witing time t us stop is uniformly distriuted on the intervl [, ]. Find the proility tht it is etween 1 nd minutes. Solution. 1 x f x;, otherwise 1 1 P1 X dx 1 1

4 STT Exmple Let X e the time hedwy for two rndomly chosen consecutive crs on freewy during period of hevy flow. The pdf of X cn e pproximted y f x.1 x..1e x. otherwise Wht is the proility tht hedwy time is t most sec? Solution. P X f x dx..1 e.1e Exmple x..1t.1t e dt 1 e dx A college professor never finishes his lecture efore the end of the hour nd lwys finishes his lectures within min fter the hour. Let the time tht elpses etween the end of the hour nd the end of the lecture nd suppose the pdf of X is kx x f x otherwise ) Find the vlue of k nd drw the corresponding density curve. Hint: Totl re under the grph of f x is 1. ) Wht is the proility tht the lecture ends within 1 min of the end of the hour? c) Wht is the proility tht the lecture continues eyond the hour for etween 6 nd 9 sec? d) Wht is the proility tht the lecture continues for t lest 9 sec eyond the end of the hour? 4

5 STT Solution. k ) kx dx 1 x 1 k 8 1 ) P X 1 x dx c) P1 X 1. x dx d) P X Numericl Chrcteristics of Continuous RVs Formuls of E X nd E h X for Continuous RVs. If X is continuous rndom vrile with pdf f f x nd h h X, then X E X x f x dx hx E h X h x f x dx A mode of continuous proility distriution is often considered to e ny vlue x t which its proility density function hs loclly mximum vlue, so ny pek is mode.

6 STT Formuls of V X nd X for continuous rv. Shortcut formul X V X X V X x f x dx ; V X E X E X E X x f x dx The vrince nd stndrd devition give quntittive mesures of how much spred there is in the distriution or popultion of x vlues. Agin is roughly the size of typicl devition from. Shortcut formul for in the cse of continuous rndom vriles is the sme s in discrete cse Computtionl exmples. 1) Uniform distriution on,. 1 x f x;, otherwise E X 1 1 x 1 xdx 1 1 x 1 E X x dx V X E X E X SD

7 STT ) Given f x cx x 1, otherwise Find the vlue of constnt c, then compute E X, V X, nd X. 1 1 x c cxdx 1 c 1 c E X E X 1 xdx 1 x dx V X E X E X 9 9 ) Given f x cos x x, /. otherwise Find E X, V X, ; EX, V X / E X xcos xdx V X E X E X / 8 E X x cos xdx 4 7

8 STT Cumultive Distriution Function of Continuous RV Ojectives. Definition of cumultive distriution function for continuous rv. Otining density from cumultive distriution function. Computing proilities y using density nd cumultive distriution functions Definition. The cumultive distriution function F for continuous rndom vrile X is defined for every numer x y x F x P X x f tdt For ech x, F(x) is the re under the density curve to the left of x. p. 144 textook 4... Connection etween f from F. If X is continuous rv with pdf f nd cdf F, then t every x t which the derivtive F exists, F x f x 8

9 STT Exmples for computing. #1. Uniform distriution on., Given pdf f for uniform distriution on., Find corresponding cdf F. 1 f x;, Solution. X otherwise F?, x 1 F dx, x ny rel numer, F, x x 1 x dt x 1 x #. Given pdf f for continuous rv X. f x x 1 x1 otherwise Find corresponding cdf F. F, x x 1 1 t dt x 1 1 x 1 1 x 1 9

10 STT Percentiles of Continuous Distriution. Let p e numer etween nd 1. The (1p)-th percentile of the distriution of continuous rv X, denoted y p, is defined y p p F p f t dt In prticulr, the medin of continuous distriution, denoted, is the th percentile, so stisfies the eqution. F. Geometric mening of lst eqution. Hlf the re under density curve is to the left of nd hlf is to the right of Exmples of density function with symmetry out some point. The medin nd the men (if it exists) of symmetric distriution oth occur t the point out which the symmetry occurs. If symmetric distriution is unimodl, the mode coincides with the medin nd men Norml Rndom Vrile Exmple of Symmetric Distriution. A continuous rv X is sid to hve norml distriution with prmeters nd (or nd ), where nd, if the pdf of X is f x x 1 e x For this distriution, the medin, mode, nd expected vlue re the sme. 1

11 STT

Chapter 6 Continuous Random Variables and Distributions

Chapter 6 Continuous Random Variables and Distributions Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Carlo Integration 02/10/05 CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes

38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes The Uniform Distribution 8. Introduction This Section introduces the simplest type of continuous probbility distribution which fetures continuous rndom vrible X with probbility density function f(x) which

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

Best Approximation in the 2-norm

Best Approximation in the 2-norm Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Best Approximation. Chapter The General Case

Best Approximation. Chapter The General Case Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

More information

Chapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses

Chapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of

More information

Numerical Integration

Numerical Integration Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture

More information

Big idea in Calculus: approximation

Big idea in Calculus: approximation Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

Continuous probability distributions

Continuous probability distributions Chpter 1 Continuous probbility distributions 1.1 Introduction We cll x continuous rndom vrible in x b if x cn tke on ny vlue in this intervl. An exmple of rndom vrible is the height of dult humn mle, selected

More information

Math Calculus with Analytic Geometry II

Math Calculus with Analytic Geometry II orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

For a continuous function f : [a; b]! R we wish to define the Riemann integral

For a continuous function f : [a; b]! R we wish to define the Riemann integral Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This

More information

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

More information

Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Least Squares Approximation Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

More information

7 - Continuous random variables

7 - Continuous random variables 7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Normal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution

Normal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution Norml Distribution Lecture 6: More Binomil Distribution If X is rndom vrible with norml distribution with men µ nd vrince σ 2, X N (µ, σ 2, then P(X = x = f (x = 1 e 1 (x µ 2 2 σ 2 σ Sttistics 104 Colin

More information

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x  0 : Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

Distance And Velocity

Distance And Velocity Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

More information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

Mapping the delta function and other Radon measures

Mapping the delta function and other Radon measures Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

Numerical integration

Numerical integration 2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

More information

3.4 Numerical integration

3.4 Numerical integration 3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,

More information

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon. EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work

More information

Discrete Least-squares Approximations

Discrete Least-squares Approximations Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

Orthogonal Polynomials and Least-Squares Approximations to Functions

Orthogonal Polynomials and Least-Squares Approximations to Functions Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

More information

Chapter 2. Random Variables and Probability Distributions

Chapter 2. Random Variables and Probability Distributions Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

More information

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests. ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

More information

Math 270A: Numerical Linear Algebra

Math 270A: Numerical Linear Algebra Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner

More information

BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

More information

dy ky, dt where proportionality constant k may be positive or negative

dy ky, dt where proportionality constant k may be positive or negative Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

Conducting Ellipsoid and Circular Disk

Conducting Ellipsoid and Circular Disk 1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,

More information

8 Laplace s Method and Local Limit Theorems

8 Laplace s Method and Local Limit Theorems 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

Euler-Maclaurin Summation Formula 1

Euler-Maclaurin Summation Formula 1 Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,

More information

COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature

COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III - Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting

More information

Chapter 6. Riemann Integral

Chapter 6. Riemann Integral Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

More information

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

MA Handout 2: Notation and Background Concepts from Analysis

MA Handout 2: Notation and Background Concepts from Analysis MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

More information

The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5 The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle

More information

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

Problem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume

Problem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom r vrible. Define new rndom vrible Y = g. Find the pdf of Y. Method: Step : Step : Step 3: Plot Y = g( ). Find F ( y) by mpping

More information

PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes. Paul Carnig. January ODE s vs PDE s 1 PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

Math 100 Review Sheet

Math 100 Review Sheet Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

More information

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

Designing Information Devices and Systems I Spring 2018 Homework 8

Designing Information Devices and Systems I Spring 2018 Homework 8 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission

More information

We know that if f is a continuous nonnegative function on the interval [a, b], then b

We know that if f is a continuous nonnegative function on the interval [a, b], then b 1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

a n+2 a n+1 M n a 2 a 1. (2)

a n+2 a n+1 M n a 2 a 1. (2) Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside

More information

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15 Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll

More information