# Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Size: px
Start display at page:

Download "Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem"

Transcription

1 Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the eal line. Suppose g maps A onto B, so that thee is an inese map x = hy) fom B back to A. Kenneth Hais Depatment of Mathematics Uniesity of Michigan Apil 3, 009 Let X be a continuous andom aiable with known density f X x). Let Y = GX). Then the density of Y is ) f Y y) = f X hy) d. dt hy) Note: Compae to Ross, Theoem 5.7., page 43. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Poblem Two Functions of Two Random Vaiables Two Functions of Two Random Vaiables Definition Nice Tansfomations Let the continuous andom aiables X, Y ) hae joint density f X,Y x, y) and let A = x, y) : f X,Y x, y) > 0}. X, Y ) detemines a point with xy-coodinates in the egion A. Conside the continuous andom aiables U, V ) gien by U = g X, Y ) V = g X, Y ). U, V ) detemines a point with u-coodinates in some egion B. Poblem. If the tansfomation fom xy-coodinates to u-coodinates gien by u = g x, y) = g x, y). is nice on A, then we can poduce the joint density f U,V u, ) fo the andom aiable U, V ). Definition A tansfomation fom xy-coodinates to u-coodinates xy u) gien by is nice on A, if u = g x, y) The patial deiaties u on A. x, u y, x = g x, y)., and y The Jacobian of the tansfomation is nonzeo on A: u u x y Jx, y) = = u x y u y x 0 whenee x, y) A. x y exist and ae continuous Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

2 Two Functions of Two Random Vaiables Change of coodinates A nice tansfomation on A xy u) amounts to simply a change of coodinates of the plane fom xy-coodinates to u-coodinates. We can ecoe the oiginal xy-coodinates fom the new u-coodinates. Suppose xy u) is nice tansfomation on A u = g x, y) = g x, y) to u-coodinates on a egion B. Thee is a eese tansfomation u xy) fom u-coodinates to xy-coodinates x = h u, ) y = h u, ). which maps B onto A and which ae also nice on B. Jacobians Two Functions of Two Random Vaiables The Jacobian of the oiginal tansfomation xy u) is the deteminant u u x y Jx, y) = = u x y u y x x The Jacobian of the inese tansfomation u xy) is the deteminant x x Ju, ) = u y y = x y u x y u Since xy u) is nice on A, Jx, y) 0 whenee x, y) A and Ju, ) 0 whenee u, ) B. u Futhemoe, the two Jacobian deteminants ae ineses y Jx, y) = Ju, ) Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Main Theoem Theoem Two Functions of Two Random Vaiables Let X, Y ) be continuous andom aiables with joint density f X,Y x, y), and U, V ) be andom aiables gien by U = g X, Y ) V = g X, Y ). Suppose the xy u) tansfomation u = g x, y) is nice on A = x, y) : f X,Y x, y) 0}. Let the inese u xy) fom B to A be x = h u, ) = g x, y). y = h u, ). Two Functions of Two Random Vaiables Pictue of Theoem f U,V u, ) du d P U, V ) B} UV P U,V B B u, P X, Y ) A} f X,Y x, y) Ju, ) du d d XY P X,Y A A x,y The joint density of U, V ) is gien fo u, ) B by eithe equation f U,V u, ) = f X,Y h u, ), h u, ) ) Ju, ) f U,V u, ) = f X,Y h u, ), h u, ) ) Jx, y) whichee is moe conenient to compute. du Aea of B du d Aea of A J u, du d Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

3 Two Functions of Two Random Vaiables Sketch of Poof of Theoem Let B B and suppose u xy) maps B to A A. P U, V ) B} = P X, Y ) A} = f X,Y x, y) dx dy = x,y) A u,) B f X,Y h u, ), h u, ) ) Ju, ) du d using the Change of Vaiables Theoem of analysis. Intuitiely, we can beak B into small egions B which u xy) tansfoms to small egions A of A whee fo any u, ) B: f U,V u, ) Aea B ) f X,Y h u, ), h u, )) Aea A ) whee Aea A ) Aea B ) Ju, ). Diffeentiate the integals to get the tansfomation ule: f X,Y h u, ), h u, ) ) Ju, ) if u, ) B f U,V u, ) = 0 othewise. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3 Functions of a Random Vaiable: Density. Let X and Y be continuous andom aiables with joint density f X,Y x, y) and whee X 0. Conside U = XY V = X. The tansfomation xy u) is gien by u = xy = x. The inese tansfomation u xy) is gien by x = y = u. The Jacobian fo tansfomation fo u xy) is Ju, ) = 0 u = Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Functions of a Random Vaiable: Density Rectangle to Pola coodinates So, the joint density is f U,V u, ) = f X,Y h u, ), h u, ) ) Ju, ) u ) = f X,Y, We can compute the maginal f U u) = f XY u) by f XY u) = u ) f X,Y, d It is often conenient to change fom ectangula coodinates xy to pola coodinates θ. The tansfomation xy θ) is whee > 0 and π < θ π. = x + y θ = actan y x. The inese tansfomation θ xy) fom pola back to ectangula is x = cos θ y = sin θ. The tansfomation is xy θ) nice in the punctued plane R 0, 0)}. Veified in thee slides. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

4 Coneting Rectangle to Pola Coneting Rectangle to Pola Rectangle xy-coodinates to pola θ-coodinates: Plot of tan y x on [ π, π]. The fou quadants of the plane ae = x + y, > 0 θ = actan y, π < θ π, x Pola θ-coodinates to ectangle xy-coodinates x = cos θ y = sin θ < x, y <. x,y,θ I : x, y > 0 II : x < 0, y > 0 III : x, y < 0 IV : x > 0, y < 0 II III 6 I IV 4 x y y sinθ Π Π Π Π 4 Θ actan y x 6 x cosθ Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Poblem: Rectangle to Pola Poblem. Let X, Y ) be andomly chosen in some egion R of the xy-plane with joint density f X,Y x, y). Conside the andom aiables giing the pola coodinates R = X + Y whee R > 0 and π < Θ π. Θ = actan Y X The Jacobian is easiest to compute on the θ-plane: J, θ) = cos θ sin θ sin θ cos θ = cos θ + sin θ = The joint distibution of R, Θ is f R,Θ, θ) = f X,Y cos θ, sin θ) > 0, π < θ π. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3. Let X, Y ) be unifomly distibuted in R = unit cicle. So, So, f X,Y x, y) = π f R,Θ, θ) = f X,Y cos θ, sin θ) = π The maginals ae f R ) = f Θ θ) = π π 0 when x + y. dθ = 0 <, π π d = π Thus, Θ is unifomly distibuted on π, π]. 0 <, π < θ π π < θ π. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

5 . Let X, Y ) be independent and nomally distibuted in the plane with µ = 0, σ ). So, f X,Y x, y) = πσ e x +y )/σ Since f R,Θ, θ) = f X,Y cos θ, sin θ), f R,Θ, θ) = πσ e cos θ+ sin θ)/σ = πσ e /σ 0 <, π < θ π. The maginals ae f R ) = f Θ θ) = π π πσ e 0 πσ e /σ dθ = σ e /σ 0 <, /σ d = π π < θ π. Thus, Θ is unifomly distibuted on π, π] and R is the Rayleigh distibution the distance of X, Y ) fom the oigin). Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3. Let R be exponentially distibuted with mean and Θ be unifomly distibuted in π, π], both independent. The joint distibution is f R,Θ, θ) = π e / 0 <, π < θ π Let X and Y be andom aiables detemined by X = R cos Θ Y = R sin Θ Sole fo, θ in the tansfomation x = cos θ and y = sin θ: = x + y θ = actan y x. The Jacobian deteminant is easiest to compute using θ-coodinates: cos θ J, θ) = sin θ = cos θ + sin θ = sin θ cos θ. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3 So, f R,Θ, θ) = π e / 0 <, π < θ π f X,Y x, y) = f R,θ x + y, actan y x ) = π e x +y )/ X and Y ae independent and nomally distibuted andom aiables with µ = 0, σ = ). The maginals ae obtained by integating f X,Y x, y): f X x) = f Y y) = π e x / π e y / continued Let U and V be unifomly distibuted on 0, ). Conside the andom aiable Θ: Θ = πv π So, Θ is unifomly distibuted on π, π). Conside the andom aiable R: R = ln U By Poposition 5.7. Ross, page 43), soling, u = e /. f R ) = f U e / ) u = e / So, R is exponentially distibuted with mean. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

6 Coneting Rectangle to Pola Let X and Y be andom aiables detemined by X = R cos Θ Y = R sin Θ Simulating a standad nomal andom aiable with a pai of independent unifom andom aiables on 0, ). Then X and Y ae independent standad nomal andom aiables!! We can simulate a standad nomal andom aiable X by using two independent unifom andom aiables U and V on 0, ): X = ln U cos πv π ). 000 data points ,000 data points Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 of Joint Distibution. Let X and Y be independent and unifomly distibuted on 0, ]. Find the joint pobability density function fo the andom aiables U = X Y V = XY. Indiidually, the distibution of X and Y ae if 0 x if 0 y f X x) = f Y y) = 0 othewise 0 othewise So, the joint distibution f X,Y x, y) is f X,Y x, y) = f X x) f Y y) = if 0 x, y 0 othewise. of Joint Distibution The tansfomation into u-coodinates u = x y is one-to-one and has an inese = xy, x = u y = u. The Jacobian deteminant is easiest when computed in xy coodinates: Jx, y) = y x y y x = x y = u So, u, > 0 and ) f U,V u, ) = f X,Y u, = u if 0 < u, u u u 0 othewise. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

7 of Joint Distibution of Joint Distibution It emains to compute the bounds on u and. 0 < u, u = 0 < u, 0 < u. Only one of these anges need be etained, depending upon whethe u 0, ] o u [, ): u if 0 < u <, 0 < u, f U,V u, ) = o, if u, 0 < u, 0 othewise. u if 0 < u <, 0 < u, f U,V u, ) = o, if u, 0 < u, 0 othewise. We compute the maginals. f U u) = f ) = u 0 u d if 0 < u < d if 0 < u < u 0 u d if u = d if u u 0 othewise 0 othewise u du if 0 < 0 othewise = ln if 0 < 0 othewise Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 of Joint Distibution Plot of aea detemined by 0 < u < = 0 < u and u = 0 < u u Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

### Model Question Paper Mathematics Class XII

Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

### Fluids Lecture 15 Notes

Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### NURBS Drawing Week 5, Lecture 10

CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

### CRRC-1 Method #1: Standard Practice for Measuring Solar Reflectance of a Flat, Opaque, and Heterogeneous Surface Using a Portable Solar Reflectometer

CRRC- Method #: Standad Pactice fo Measuing Sola Reflectance of a Flat, Opaque, and Heteogeneous Suface Using a Potable Sola Reflectomete Scope This standad pactice coves a technique fo estimating the

### Saturated and weakly saturated hypergraphs

Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 6-7 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B

### PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

### 1. (from Stewart, page 586) Solve the initial value problem.

. (from Stewart, page 586) Solve the initial value problem.. (from Stewart, page 586) (a) Solve y = y. du dt = t + sec t u (b) Solve y = y, y(0) = 0., u(0) = 5. (c) Solve y = y, y(0) = if possible. 3.

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

### Uniform Rectilinear Motion

Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 11-1 Engineeing Mechanics : Dynamics

### Multiple choice questions [60 points]

1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### MULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then

MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

### Week 3-4: Permutations and Combinations

Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

### Lecture 8. Generating a non-uniform probability distribution

Discrete outcomes Lecture 8 Generating a non-uniform probability distribution Last week we discussed generating a non-uniform probability distribution for the case of finite discrete outcomes. An algorithm

### Change of Variables in Double Integrals

Change of Variables in Double Integrals Part : Area of the Image of a egion It is often advantageous to evaluate (x; y) da in a coordinate system other than the xy-coordinate system. In this section, we

### Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

### 12.1. FÖRSTER RESONANCE ENERGY TRANSFER

ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to

### The Binomial Distribution

The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

### AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

### Supplementary Material for EpiDiff

Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module

### Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

### How To Find The Optimal Stategy For Buying Life Insuance

Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

### Nontrivial lower bounds for the least common multiple of some finite sequences of integers

J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.

PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### Chapter 30: Magnetic Fields Due to Currents

d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie.

### A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

### Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo

### LINES AND PLANES IN R 3

LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.

### Effect of Contention Window on the Performance of IEEE 802.11 WLANs

Effect of Contention Window on the Pefomance of IEEE 82.11 WLANs Yunli Chen and Dhama P. Agawal Cente fo Distibuted and Mobile Computing, Depatment of ECECS Univesity of Cincinnati, OH 45221-3 {ychen,

### An Efficient Group Key Agreement Protocol for Ad hoc Networks

An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

### ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

### Experiment 6: Centripetal Force

Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

### Feb 28 Homework Solutions Math 151, Winter 2012. Chapter 6 Problems (pages 287-291)

Feb 8 Homework Solutions Math 5, Winter Chapter 6 Problems (pages 87-9) Problem 6 bin of 5 transistors is known to contain that are defective. The transistors are to be tested, one at a time, until the

### MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

### YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

### 4.1 - Trigonometric Functions of Acute Angles

4.1 - Tigonometic Functions of cute ngles a is a half-line that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one

### Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

### An Epidemic Model of Mobile Phone Virus

An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity

### AB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss

AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =

### 10 Polar Coordinates, Parametric Equations

Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates

### Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w

1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a

### Solutions to Practice Problems for Test 4

olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,

### Introduction to Fluid Mechanics

Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body

### Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

### dz + η 1 r r 2 + c 1 ln r + c 2 subject to the boundary conditions of no-slip side walls and finite force over the fluid length u z at r = 0

Poiseuille Flow Jean Louis Maie Poiseuille, a Fench physicist and physiologist, was inteested in human blood flow and aound 1840 he expeimentally deived a law fo flow though cylindical pipes. It s extemely

### Math Placement Test Practice Problems

Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

### A discus thrower spins around in a circle one and a half times, then releases the discus. The discus forms a path tangent to the circle.

Page 1 of 6 11.2 Popeties of Tangents Goal Use popeties of a tangent to a cicle. Key Wods point of tangency p. 589 pependicula p. 108 tangent segment discus thowe spins aound in a cicle one and a half

### Area and Arc Length in Polar Coordinates

Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly

### Chapter 4: Fluid Kinematics

Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

### AP Physics Electromagnetic Wrap Up

AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle

### The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit

### An Analysis of Manufacturer Benefits under Vendor Managed Systems

An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1

### MATH 381 HOMEWORK 2 SOLUTIONS

MATH 38 HOMEWORK SOLUTIONS Question (p.86 #8). If g(x)[e y e y ] is harmonic, g() =,g () =, find g(x). Let f(x, y) = g(x)[e y e y ].Then Since f(x, y) is harmonic, f + f = and we require x y f x = g (x)[e

### Calculus with Parametric Curves

Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function

### Definitions. Optimization of online direct marketing efforts. Test 1: Two Email campaigns. Raw Results. Xavier Drèze André Bonfrer. Lucid.

Definitions Optimization of online diect maketing effots Xavie Dèze Andé Bonfe Lucid Easily undestood; intelligible. Mentally sound; sane o ational. Tanslucent o tanspaent. Limpid Chaacteized by tanspaent

### Seshadri constants and surfaces of minimal degree

Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### Solutions - Homework sections 17.7-17.9

olutions - Homework sections 7.7-7.9 7.7 6. valuate xy d, where is the triangle with vertices (,, ), (,, ), and (,, ). The three points - and therefore the triangle between them - are on the plane x +

### Principle of Data Reduction

Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then

### Chapter 2. Electrostatics

Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### Symmetric polynomials and partitions Eugene Mukhin

Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation

### Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

### Experimentation under Uninsurable Idiosyncratic Risk: An Application to Entrepreneurial Survival

Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival Jianjun Miao and Neng Wang May 28, 2007 Abstact We popose an analytically tactable continuous-time model of expeimentation

### It is required to solve the heat-condition equation for the excess-temperature function:

Jounal of Engineeing Physics and Themophysics. Vol. 73. No. 5. 2 METHOD OF PAIED INTEGAL EQUATIONS WITH L-PAAMETE IN POBLEMS OF NONSTATIONAY HEAT CONDUCTION WITH MIXED BOUNDAY CONDITIONS FO AN INFINITE

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS ` E MISN-0-133. CHARGE DISTRIBUTIONS by Peter Signell, Michigan State University

MISN-0-133 GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS by Pete Signell, Michigan State Univesity 1. Intoduction..............................................

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

### Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

### MATHEMATICAL SIMULATION OF MASS SPECTRUM

MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec

### Secure Smartcard-Based Fingerprint Authentication

Secue Smatcad-Based Fingepint Authentication [full vesion] T. Chales Clancy Compute Science Univesity of Mayland, College Pak tcc@umd.edu Nega Kiyavash, Dennis J. Lin Electical and Compute Engineeing Univesity

### The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the