Joint distribution. Joint distribution. Marginal distributions. Joint distribution
|
|
- Amberlynn Long
- 5 years ago
- Views:
Transcription
1 Joint distribution To specify the joint distribution of n rndom vribles X 1,...,X n tht tke vlues in the smple spces E 1,...,E n we need probbility mesure, P, on E 1... E n = {(x 1,...,x n ) x i E i, i = 1,...,n}. Joint distribution To specify the joint distribution of n rndom vribles X 1,...,X n tht tke vlues in the smple spces E 1,...,E n we need probbility mesure, P, on E 1... E n = {(x 1,...,x n ) x i E i, i = 1,...,n}. For A E 1... E n P((X 1,...,X n ) A) = P(A) nd for A 1 E 1,...,A n E n P(X 1 A 1,...,X n A n ) = P(A 1... A n ). We sy tht P is the joint distribution of the bundled vrible X =. p.1/3 (X 1,...,X n ).. p.1/3 Joint distribution Mrginl distributions To specify the joint distribution of n rndom vribles X 1,...,X n tht tke vlues in the smple spces E 1,...,E n we need probbility mesure, P, on E 1... E n = {(x 1,...,x n ) x i E i, i = 1,...,n}. For A E 1... E n If P is the joint distribution of X = (X 1,...,X n ) we cn get the mrginl distribution of X i s P i (A) = P(X i A) = P(E 1... E i 1 A E i+1... E n ). for A E i. P((X 1,...,X n ) A) = P(A) nd for A 1 E 1,...,A n E n P(X 1 A 1,...,X n A n ) = P(A 1... A n ).. p.1/3. p./3
2 Mrginl distributions Independence If P is the joint distribution of X = (X 1,...,X n ) we cn get the mrginl distribution of X i s P i (A) = P(X i A) = P(E 1... E i 1 A E i+1... E n ). for A E i. Specifiction of the mrginl distributions lone is not enough to specify the joint distribution we lso need to specify how the vribles we consider re relted. Definition: We sy tht X 1,...,X n re independent if P(X 1 A 1,...,X n A n ) = P(X i A 1 )... P(X n A n ). (1) If we specify the mrginl distributions of X 1,...,X n nd sy tht the vribles re independent then we hve specified the joint distribution by eqution (1).. p./3. p.3/3 Independence Trnsformtions Definition: We sy tht X 1,...,X n re independent if P(X 1 A 1,...,X n A n ) = P(X i A 1 )... P(X n A n ). (1) Theorem: If X 1 nd X re independent rndom vribles tking vlues in E 1 nd E respectively, nd if h 1 : E 1 E 1 nd h : E E re two trnsformtions then the rndom vribles h 1 (X 1 ) nd h (X ) re independent. Mrginl trnsformtions preserve independence.. p.3/3. p.4/3
3 Discrete smple spces Exmple If E 1,...,E n re discrete smple spces so is E = E 1...E n nd the joint distribution is given in terms of point probbilities p(x 1,...,x n ), x i E i, i = 1,...,n. Consider E = E 0 E 0 = {A, C, G, T} {A, C, G, T}, nd we let X nd Y denote rndom vribles representing two evolutionry relted nucleic cids in DNA sequence. Let the joint distribution of X nd Y hve point probbilities A C G T p.5/3. p.6/3 Discrete smple spces Exmple If E 1,...,E n re discrete smple spces so is E = E 1...E n nd the joint distribution is given in terms of point probbilities p(x 1,...,x n ), x i E i, i = 1,...,n. The mrginl distribution of X i hs point probbilities p i (x i ) = P(X i = x i ) = for x i E i. x 1,...,x i 1,x i+1,...,x n p(x 1,...,x i 1, x, x i+1,...,x n ) Consider E = E 0 E 0 = {A, C, G, T} {A, C, G, T}, nd we let X nd Y denote rndom vribles representing two evolutionry relted nucleic cids in DNA sequence. Let the joint distribution of X nd Y hve point probbilities A C G T Let A = {(x, y) E x = y} denote the event tht the two relted nucleic cids re identicl then. p.5/3 P(X = Y ) = P(A) = = p.6/3
4 Independence nd point probbilities Exmple Theorem: If X 1,...,X n re rndom vribles with vlues in discrete smple spces then they re independent if nd only if P(X 1 = x 1,...,X n = x n ) = P(X 1 = x 1 )... P(X n = x 1 ) X Y A C G T Sme exmple s bove but with the point probbilities for the mrginl distributions. Note tht X nd Y re not independent! For instnce 0.17 = P((X, Y ) = (A, A)) P(X = A) P(Y = A) = = p.7/3. p.8/3 Independence nd point probbilities Exmple Theorem: If X 1,...,X n re rndom vribles with vlues in discrete smple spces then they re independent if nd only if P(X 1 = x 1,...,X n = x n ) = P(X 1 = x 1 )... P(X n = x 1 ) In words, the rndom vribles re independent if nd only if the point probbilities for their joint distribution fctorize s product of the point probbilities for their mrginl distributions. X Y A C G T Sme mrginls s bove but X nd Y re independent in this exmple.. p.7/3. p.9/3
5 The bivrite norml distribution The bivrite norml distribution The function f(x, y) = 1 ρ exp ( x ρxy + y ) π for ρ ( 1, 1) on R is n exmple of bivrite density on R. It stisfies tht f(x) 0 nd f(x, y)dxdy = 1 density x y density x y With ρ = 0 (left) nd ρ = 0.75 (right).. p.10/3. p.11/3 The bivrite norml distribution The bivrite norml distribution The function f(x, y) = 1 ρ exp ( x ρxy + y ) π We find tht the mrginl distribution of X is given by P( X b) = b f(x, y)dydx = b (1 ρ ) π e (1 ρ )x dx, for ρ ( 1, 1) on R is n exmple of bivrite density on R. It stisfies tht f(x) 0 nd f(x, y)dxdy = 1 The rndom vribles X nd Y hve joint distribution with density f if P( 1 X b 1, Y b ) = b1 b 1 f(x, y)dxdy. p.10/3. p.1/3
6 The bivrite norml distribution The bivrite norml distribution We find tht the mrginl distribution of X is given by P( X b) = b f(x, y)dydx = b (1 ρ ) π e (1 ρ )x dx, We find tht the mrginl distribution of X is given by P( X b) = b f(x, y)dydx = b (1 ρ ) π e (1 ρ )x dx, which shows tht the mrginl distribution of X is N(0, (1 ρ ) 1 ). which shows tht the mrginl distribution of X is N(0, (1 ρ ) 1 ). The mrginl distribution of Y is N(0, (1 ρ ) 1 ), but the ρ lso determines dependence between X nd Y. The vribles X nd Y re independent if nd only if ρ = 0.. p.1/3. p.1/3 The bivrite norml distribution We find tht the mrginl distribution of X is given by P( X b) = b f(x, y)dydx = b (1 ρ ) π e (1 ρ )x dx, which shows tht the mrginl distribution of X is N(0, (1 ρ ) 1 ). Generl results If X 1,...,X n re rel vlued rndom vribles we cn specify their joint distribution by density f : R n [0, ) such tht P( 1 X 1 b 1,..., n X n b n ) = b1 1 bn n f(x 1,...,x n )dx n...dx 1 cn be computed s n successive ordinry integrls (order does not mtter). The mrginl distribution of Y is N(0, (1 ρ ) 1 ), but the ρ lso determines dependence between X nd Y.. p.1/3. p.13/3
7 Generl results The model builders pproch If X 1,...,X n re rel vlued rndom vribles we cn specify their joint distribution by density f : R n [0, ) such tht P( 1 X 1 b 1,..., n X n b n ) = b1 1 bn n f(x 1,...,x n )dx n...dx 1 We wnt to construct probbilistic model for the rndom vribles X 1,...,X n. cn be computed s n successive ordinry integrls (order does not mtter). The mrginl distribution of X i hs density f i (x i ) = f(x 1,...,x n )dx n...dx i+1 dx i 1...dx 1. }{{} n 1. p.13/3. p.14/3 Generl results The model builders pproch If X 1,...,X n re rel vlued rndom vribles we cn specify their joint distribution by density f : R n [0, ) such tht P( 1 X 1 b 1,..., n X n b n ) = b1 1 bn n f(x 1,...,x n )dx n...dx 1 We wnt to construct probbilistic model for the rndom vribles X 1,...,X n. Cn we ssume tht the vribles re independent if yes continue. cn be computed s n successive ordinry integrls (order does not mtter). The mrginl distribution of X i hs density f i (x i ) = f(x 1,...,x n )dx n...dx i+1 dx i 1...dx 1. }{{} n 1 The X i s re independent if f(x 1,...,x n ) = f 1 (x 1 )... f n (x n ).. p.13/3. p.14/3
8 The model builders pproch Conditionl distributions We wnt to construct probbilistic model for the rndom vribles X 1,...,X n. Cn we ssume tht the vribles re independent if yes continue. Cn we ssume tht the vribles ll hve the sme mrginl distribution if yes continue. Definition: The conditionl distribution of Y given tht X A is defined s P(Y B X A) = provided tht P(X A) > 0. P(Y B, X A) P(X A). p.14/3. p.15/3 The model builders pproch Conditionl distributions We wnt to construct probbilistic model for the rndom vribles X 1,...,X n. Cn we ssume tht the vribles re independent if yes continue. Cn we ssume tht the vribles ll hve the sme mrginl distribution if yes continue. Then we sy: Let X 1,...,X n be iid = independent nd identiclly distributed. Definition: The conditionl distribution of Y given tht X A is defined s P(Y B X A) = provided tht P(X A) > 0. P(Y B, X A) P(X A) If X nd Y re discrete we cn condition on events X = x nd get conditionl distributions in terms of point probbilities And we need to specify either the point probbilities for their common distribution or the density the joint distribution is given by products. p(y x) = P(Y = y X = x) = P(Y = y, X = x) P(X = x) where p(x, y) re the joint point probbilities. = p(x, y) p(x, y) y. p.14/3. p.15/3
9 Exmple Systemtic specifiction Using P(Y = y X = x) = P(X = x, Y = y) P(X = x) = p(x, y) p(x, y). y E we hve to divide by precisely the row sums to get the mtrix of conditionl distributions: X Y A C G T The row sums bove equl 1 nd this is n exmple of mtrix of trnsition Dependence mong discrete vribles indexed by time prmeter cn be treted systemticlly. We cn define collection of conditionl probbilities, P t (x, y) for t 0, of Y = y given X = x s the solution to system of differentil equtions: dp t (x, y) dt for λ(z, y) 0 for z y nd The λ(y, z) s re clled intensities. = z P t (x, z)λ(z, y) λ(z, z) = y z λ(y, z). probbilities.. p.16/3. p.17/3 Systemtic specifiction Solution Dependence mong discrete vribles indexed by time prmeter cn be treted systemticlly. We cn define collection of conditionl probbilities, P t (x, y) for t 0, of Y = y given X = x s the solution to system of differentil equtions: On finite smple spce nd with the initil condition P 0 (x, x) = 1 the bove system of differentil equtions hs unique solution such tht P t (x, ) is (conditionl) probbility mesure for ll x. dp t (x, y) dt = z P t (x, z)λ(z, y) In generl no closed form expression for the solution. for λ(z, y) 0 for z y nd λ(z, z) = y z λ(y, z).. p.17/3. p.18/3
10 Jukes-Cntor model Liner Regression Intensities: A 3α α α α C α 3α α α G α α 3α α T α α α 3α We often specify the conditionl distribution of rel vlued rndom vrible Y given X = x for nother rel vlued rndom vrible X by writing Y = α + βx + ǫ where ǫ is nother men 0 rndom vrible (noise), which is independent of X. The prmeter α > 0 tells how mny muttions tht occur per time unit. The solution is P t (x, x) = exp( 4αt) P t (x, y) = exp( 4αt), if x y,. p.19/3. p.1/3 Kimur model Liner Regression Intensities: A α β β α β C β α β β α G α β α β β T β α β α β for α, β > 0 nd the solution is We often specify the conditionl distribution of rel vlued rndom vrible Y given X = x for nother rel vlued rndom vrible X by writing Y = α + βx + ǫ where ǫ is nother men 0 rndom vrible (noise), which is independent of X. This is loction trnsformtion of the distribution of ǫ. The conditionl men of Y given X = x is α + βx. P t (x, x) = exp( 4βt) exp( (α + β)t) P t (x, y) = exp( 4βt) 0.5 exp( (α + β)t), if λ(x, y) = α P t (x, y) = exp( 4βt), if λ(x, y) = β,. p.0/3. p.1/3
11 Liner Regression Conditionl densities We often specify the conditionl distribution of rel vlued rndom vrible Y given X = x for nother rel vlued rndom vrible X by writing Y = α + βx + ǫ where ǫ is nother men 0 rndom vrible (noise), which is independent of X. This is loction trnsformtion of the distribution of ǫ. The conditionl men of Y given X = x is α + βx. Definition: If f is the density for the joint distribution of two rndom vribles X nd Y tking vlues in R n nd R m, respectively, then with f 1 (x) = f(x, y)dy R m we define the conditionl distribution of Y given X = x to be the distribution with density f(x, y) f(y x) = f 1 (x). for y R m nd x R n with f 1 (x) > 0. If ǫ N(0, σ ) then Y X = x N(α + βx, σ ). Note the formul f(x, y) = f(y x)f 1 (x), tht llows us to specify the joint deistribution by specifying the mrginl. p.1/3 distribution of X nd the conditionl distribution of Y given X.. p./3 Conditionl densities Generlized liner models Definition: If f is the density for the joint distribution of two rndom vribles X nd Y tking vlues in R n nd R m, respectively, then with f 1 (x) = f(x, y)dy R m we define the conditionl distribution of Y given X = x to be the distribution with density f(x, y) f(y x) = f 1 (x). We consider the setup with the probbility mesures P θ on discrete E 1 R given by the point probbilities for θ Θ R. p θ (x) = exp(θx b(θ) + c(x)) for y R m nd x R n with f 1 (x) > 0.. p./3. p.3/3
12 Generlized liner models We consider the setup with the probbility mesures P θ on discrete E 1 R given by the point probbilities for θ Θ R. p θ (x) = exp(θx b(θ) + c(x)) Let Y be rel vlued rndom vrible we cn define the conditionl distribution of X given Y = y to be P β0 +β 1 y.. p.3/3 Generlized liner models We consider the setup with the probbility mesures P θ on discrete E 1 R given by the point probbilities for θ Θ R. p θ (x) = exp(θx b(θ) + c(x)) Let Y be rel vlued rndom vrible we cn define the conditionl distribution of X given Y = y to be P β0 +β 1 y. Tht is, the conditionl point probbilites for the distribution of X given Y = y re p(x y) = p β0 +β 1 y(x) = exp((β 0 + β 1 y)x b(β 0 + β 1 y) + c(x)). The conditionl men of X given Y = y is b (β 0 + β 1 y).. p.3/3
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 informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationMethod: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom vrible. Define new rndom vrible Y g( ) ). Find the pdf of Y. Method: Step : Step : Step 3: Plot Y g( ). Find F ( ) b mpping
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More information7 - 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 informationX Z Y Table 1: Possibles values for Y = XZ. 1, p
ECE 534: Elements of Informtion Theory, Fll 00 Homework 7 Solutions ll by Kenneth Plcio Bus October 4, 00. Problem 7.3. Binry multiplier chnnel () Consider the chnnel Y = XZ, where X nd Z re independent
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationTutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits.
Tutoril 4 Exercises on Differentil Entropy. Evlute the differentil entropy h(x) f ln f for the following: () The uniform distribution, f(x) b. (b) The exponentil density, f(x) λe λx, x 0. (c) The Lplce
More informationProblem. 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 informationLecture 21: Order statistics
Lecture : Order sttistics Suppose we hve N mesurements of sclr, x i =, N Tke ll mesurements nd sort them into scending order x x x 3 x N Define the mesured running integrl S N (x) = 0 for x < x = i/n for
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationCS667 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(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationSection 17.2 Line Integrals
Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We
More informationMath 115 ( ) Yum-Tong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx
Mth 5 2006-2007) Yum-Tong Siu Lgrnge Multipliers nd Vritionl Problems with Constrints Integrl Constrints. Consider the vritionl problem of finding the extremls for the functionl J[y] = F x,y,y )dx with
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationdf dt f () b f () a dt
Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationSolution for Assignment 1 : Intro to Probability and Statistics, PAC learning
Solution for Assignment 1 : Intro to Probbility nd Sttistics, PAC lerning 10-701/15-781: Mchine Lerning (Fll 004) Due: Sept. 30th 004, Thursdy, Strt of clss Question 1. Bsic Probbility ( 18 pts) 1.1 (
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationFunctions of Several Variables
Functions of Severl Vribles Sketching Level Curves Sections Prtil Derivtives on every open set on which f nd the prtils, 2 f y = 2 f y re continuous. Norml Vector x, y, 2 f y, 2 f y n = ± (x 0,y 0) (x
More information5 Probability densities
5 Probbility densities 5. Continuous rndom vribles 5. The norml distribution 5.3 The norml pproimtion to the binomil distribution 5.5 The uniorm distribution 5. Joint distribution discrete nd continuous
More informationIntegral equations, eigenvalue, function interpolation
Integrl equtions, eigenvlue, function interpoltion Mrcin Chrząszcz mchrzsz@cernch Monte Crlo methods, 26 My, 2016 1 / Mrcin Chrząszcz (Universität Zürich) Integrl equtions, eigenvlue, function interpoltion
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd
More informationMath 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 information12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS
1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility
More informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the
More information4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for.
4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX Some reliminries: Let A be rel symmetric mtrix. Let Cos θ ; (where we choose θ π for Cos θ 4 purposes of convergence of the scheme)
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More information3.4 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 informationMATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.
MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationA Matrix Algebra Primer
A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,
More informationBest 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 informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationHW3 : Moment functions Solutions
STAT/MATH 395 A - PROBABILITY II UW Spring Qurter 6 Néhémy Lim HW3 : Moment functions Solutions Problem. Let X be rel-vlued rndom vrible on probbility spce (Ω, A, P) with moment generting function M X.
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More informationBest 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 informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationJEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS
JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODE-D. Let, b nd c be three non-zero vectors such tht no two of them re colliner nd, b c b c. If is the ngle
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationLinearity, linear operators, and self adjoint eigenvalue problems
Linerity, liner opertors, nd self djoint eigenvlue problems 1 Elements of liner lgebr The study of liner prtil differentil equtions utilizes, unsurprisingly, mny concepts from liner lgebr nd liner ordinry
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationVariational Techniques for Sturm-Liouville Eigenvalue Problems
Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationPartial 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 informationMapping 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 information38 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 informationDecision Networks. CS 188: Artificial Intelligence Fall Example: Decision Networks. Decision Networks. Decisions as Outcome Trees
CS 188: Artificil Intelligence Fll 2011 Decision Networks ME: choose the ction which mximizes the expected utility given the evidence mbrell Lecture 17: Decision Digrms 10/27/2011 Cn directly opertionlize
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationExpectation and Variance
Expecttion nd Vrince : sum of two die rolls P(= P(= = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 P(=2) = 1/36 P(=3) = 1/18 P(=4) = 1/12 P(=5) = 1/9 P(=7) = 1/6 P(=13) =? 2 1/36 3 1/18 4 1/12 5 1/9 6 5/36 7 1/6
More informationMath 231E, Lecture 33. Parametric Calculus
Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider
More informationMath Lecture 23
Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More information20 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 informationSTURM-LIOUVILLE THEORY, VARIATIONAL APPROACH
STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationLecture 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 informationProbability Distributions for Gradient Directions in Uncertain 3D Scalar Fields
Technicl Report 7.8. Technische Universität München Probbility Distributions for Grdient Directions in Uncertin 3D Sclr Fields Tobis Pfffelmoser, Mihel Mihi, nd Rüdiger Westermnn Computer Grphics nd Visuliztion
More informationx = 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 informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationMath 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 informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationMORE 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 informationNumerical 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 informationSection 3.2 Maximum Principle and Uniqueness
Section 3. Mximum Principle nd Uniqueness Let u (x; y) e smooth solution in. Then the mximum vlue exists nd is nite. (x ; y ) ; i.e., M mx fu (x; y) j (x; y) in g Furthermore, this vlue cn e otined y point
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More information1.3 The Lemma of DuBois-Reymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationJim Lambers MAT 280 Spring Semester Lecture 26 and 27 Notes
Jim Lmbers MAT 280 pring emester 2009-10 Lecture 26 nd 27 Notes These notes correspond to ection 8.6 in Mrsden nd Tromb. ifferentil Forms To dte, we hve lerned the following theorems concerning the evlution
More information1 2-D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2-D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More information38.2. 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 informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationSTURM-LIOUVILLE BOUNDARY VALUE PROBLEMS
STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2
More informationPHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS
PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?
More informationTHIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions
THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, Hui-Hsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.
More informationMath 32B Discussion Session Session 7 Notes August 28, 2018
Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationSection 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 informationSturm-Liouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1
Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition
More information