X Z Y Table 1: Possibles values for Y = XZ. 1, p
|
|
- Lorin Floyd
- 5 years ago
- Views:
Transcription
1 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 binry rndom vribles tht tke on vlues 0 nd. Z is Bernoulli(α) [i.e., P (Z = ) = α]. Find the cpcity of this chnnel nd the mximizing distribution on X. Define the distribution of X s follows: X Z Y Tble : Possibles vlues for Y = XZ. p(x) = { 0, p, p Then, we hve the following distribution for Y: { 0, pα p(y) =, pα So, we cn compute the cpcity s: C = mx dist I(X; Y ) () = H(Y ) H(Y X) () = H(pα) [P [X = 0]H(Y X = 0) + P [X = ]H(Y X = )] (3) = H(pα) ph(α) (4) Since, H(Y X = 0) = 0 nd H(Y X = ) = H(Z) = H(α).
2 Now, we need to find the prmeter p which mximizes the mutul informtion. We tke the pproch of the derivtive equled to zero. C = H(pα) ph(α) (5) = pα log pα ( pα) log ( pα) ph(α) (6) αp We obtin: 0 = d dp ( pα log pα ( pα) log ( pα) ph(α)) (7) = α log pα α log (e) + α log ( pα) + α log (e) H(α) (8) = α log pα + α log ( pα) H(α) (9) H(α) = α (log pα log ( pα)) (0) H(α) pα log α = ( ( pα) ) () pα = ( pα) H(α) α () ) ( + H(α) α = H(α) α (3) p = = H(α) α ( + H(α) α )α ( H(α) α + )α (4) (5) Then, we cn compute the cpcity: C = H(pα) ph(α) (6) ( ) = H H(α) (7) ( H(α) α + ) ( H(α) α + )α (b) Now suppose tht the receiver cn observe Z s well s Y. Wht is the cpcity? If we observe Z nd Y, the expression for the cpcity is: C = mx dist I(X; Y, Z) (8) I(X; Z) = 0 since they re independent. I(X; Y, Z) = I(X; Z) + I(X; Y Z) (9) I(X; Y Z) = H(Y Z) H(Y X, Z) (0)
3 H(Y X, Z) = 0 since given X nd Z, there is no uncertinty in Y. I(X; Y Z) = H(Y Z) () = P (Z = 0)H(Y Z = 0) + P (Z = )H(Y Z = ) () = P (Z = )H(Y Z = ) (3) = αh(x) (4) = αh(p) (5) Then the cpcity: C = mx dist I(X; Y, Z) (6) = mx dist αh(p) (7) = α (8) Since, the distribution tht mximizes the H(p) is obtined for p = /. 3
4 . Problem 7.8. Choice of chnnels. Find the cpcity C of the union of two chnnels (X, p (y x ), Y ) nd (X, p(y x ), Y ), where t ech time, one cn send symbol over chnnel or chnnel but not both. Assume tht the output lphbets re distinct nd do not intersect. () Show tht C = C + C with cpcity C.. Thus, C is the effective lphbet size of chnnel Solution: In this communiction system we cn choose between two sub-chnnels with certin probbility, lets cll it λ. We cn define Bernoulli(λ) rndom vrible: Q = {, use sub-chnnel with probbility λ, use sub-chnnel with probbility λ So, we cn see the input of the chnnel s X = (Q, X Q ). We lso hve tht since Y nd Y don t intersect, Q = f(y ), so we cn do: I(X; Y, Q) = I(X Q, Q; Y, Q) (9) = I(Y, Q; Q) + I(Y, Q; X Q Q) (30) = I(Q; Q) + I(Q; Y Q) + I(Y ; X Q Q) (3) = H(Q) H(Q Q) + H(Q Q) H(Q Y, Q) + I(Y ; X Q Q) (3) = H(Q) + I(Y ; X Q Q) (33) = H(λ) + λi(y ; X Q Q = ) + ( λ)i(y ; X Q Q = ) (34) = H(λ) + λ(y ; X ) + ( λ)i(y ; X ) (35) Cpcity follows from the mutul informtion: We cn find the λ which mximizes the cpcity: We obtin λ: C = mx dist:λ λc + ( λ)c + H(λ) (36) 0 = d dλ (λc + ( λ)c + H(λ)) (37) = C C + d dλ ( λ log (λ) ( λ) log ( λ)) (38) = C C log (λ) + log ( λ) (39) λ C C = log ( λ ) (40) C C = λ λ (4) λ = C C + C C (4) = + C C (43) 4
5 Now we cn compute the cpcity: C = λc + ( λ)c + H(λ) (44) = λc + C λc + H(λ) (45) = λ(c C ) + C λ log (λ) ( λ) log ( λ) (46) = (C C ) + C C + log ( ( + C C ) C C ) (log + C C + + C C ( + C C ) log C C ) + C = (C C ) + C C + log ( + C C C C ) (C C ) + C C + C (48) = (C C ) + log ( + C C ) + C (49) C = C ( C C + ) (50) = C + C So we cn conclude tht: (47) (5) C = log ( C + C ) (5) (b) Compre with Problem.0 where H = H + H, nd interpret prt () in terms of the effective number of noise-free symbols. From prt (), we know tht C is the effective lphbet size of chnnel with cpcity C, where effective mens the noiseless symbols, then from the expression we cn deduce tht C nd C relte to the sme concept for their corresponding sub-chnnels. (c) Use the bove result to clculte the cpcity of the following chnnel. Here we hve BSC nd noiseless binry chnnel. So, we hve C = H(p) bits, nd C = bit. 5
6 Hence: C = log ( C + C ) (53) = log ( H(p) + ) (54) 3. Problem 7.3. Source nd chnnel. We wish to encode Bernoulli(α) process V, V,... for trnsmission over binry symmetric chnnel with crossover probbility p. Find conditions on α nd p so tht the probbility of error P ( ˆV n V n ) cn be mde to go to zero s n. Solution: For Bernoulli process, we hve sequence of binry rndom vribles which re identicl nd independent. Given Eq in textbook, we know tht we cn trnsmit sttionry ergodic source over chnnel if n only if tis entropy rte is less thn the cpcity of the chnnel : H(V) < C (55) So, we compute the entropy rte for V, V,... i.i.d. rndom vribles: H(V) = lim n n H(V, V,..., V n ) (56) = lim n n nh(v ) (57) = H(V ) = H(α) (58) Now, considering tht we hve BSC with cpcity C BSC = H(p) we obtin: H(V) < C BSC (59) H(α) < H(p) (60) 6
7 4. Problem 8.. Differentil entropy. Evlute the differentil entropy h(x) = f ln f for the following: () The exponentil density, f(x) = λe λx, x 0. h(x) = = 0 0 λe λx ln (λe λx )dx (6) λe λx (ln (λ) λx)dx (6) = ln (λ) λe λx dx + λ xe λx dx (63) 0 0 ( ) ( = ln (λ)(e λx ) ( + λx)e λx) 0 0 (64) = ln (λ) + (65) This result is given in nts, since we re using the nturl logrithm. It cn be converted into bits s following: H b (X) = (log b )H (X). (b) The Lplce density, f(x) = λe λ x. h(x) = ln (λ) + (66) = log (e)( ln (λ)) (67) = log (e) log (e) ln (λ)) (68) = log (e) log (e) log (λ) log (e) (69) = log (e) log (λ) (70) h(x) = λe λ x ln ( λe λ x )dx (7) = λe λ x (ln ( λ) λ x )dx (7) = λe λ x (ln ( λ) λ x )dx (73) = λe λ x ln ( λ)dx + λe λ x λ x dx (74) = ln ( λ) λe λ x dx + λ x e λ x dx (75) 7
8 Solving the integrls: λe λ x dx = 0 ( e λx) 0 λe λ( x) dx + 0 e λx) λe λ(x) dx (76) ( = + (77) 0 = = (78) Then, λ x e λ x dx = 0 h(x) = ln ( λ) λ ( x)e λx dx + = (( λx)e (λx)) 0 0 λ (x)e λx dx (79) (( + λx)e ( λx)) 0 (80) = + = (8) λe λ x dx + λ x e λ x dx (8) = ln ( λ)( ) + () (83) = ln ( λ) + (84) As in prt (), this result is in nts, we cn convert this results into bits using the sme procedure s before: h(x) = ln ( λ) (85) = log (e)( ln ( λ)) (86) = log (e) log (e) ln ( λ)) (87) = log (e) log (e) log ( λ) log (e) (88) = log (e) log ( λ) (89) (c) The sum of X nd X, where X nd X re independent norml rndom vribles with mens µ i nd vrinces σi, i =,. We hve vribles: X N(µ, σ ) nd X N(µ, σ ). Then we obtin the Gussin rndom vrible: X + X N(µ + µ, σ + σ ) 8
9 The differentil entropy for Gussin rndom vrible is given by: h(x + X ) = log (πe(σ + σ )) Since, the men does not ffect the distribution of Gussin rndom vrible. 5. Problem 8.3. Uniformly distributed noise. Let the input rndom vrible X to chnnel be uniformly distributed over the intervl / x /. Let the output of the chnnel be Y = X + Z, where the noise rndom vrible is uniformly distributed over the intervl / z +/. () Find I(X; Y ) s function of. First we hve tht H(Y X) = H(Z) = ln (). I(X; Y ) = H(Y ) H(Y X) (90) = H(Y ) H(Z) (9) Then, we need to compute H(Y). Since Y = X + Z, we know tht the distribution of the sum of two rndom vribles is given by the convolution of their pdfs. For < we hve: f Y (y) = (+) (y +, ), (+) ( y + ), (+) y ( ) ( ) y ( ) ( ) y (+) So, to compute H(Y ), we cn observe the pdf nd see tht it cn be divided in two prts, one corresponds to uniform distribution for given probbility (i.e. λ) nd two smll tringles tht form tringulr distribution with probbility ( λ). From this observtion we cn lter use tbles (for instnce: entropy) to compute the differentil entropy of prticulr continuous distribution. 9
10 So, we cn see Y s two disjoint rndom vribles Y nd Y which hppen to be Y depending on certin probbility. Y cn be ssigned to the uniform prt, nd Y to the tringulr prt of the totl distribution. { Y, with probbility λ Y = Y, with probbility λ The next step is to define Bernoulli(λ) rndom vrible θ = f(y ) which comes from the behvior of rndom vrible Y s follows: {, if Y = Y θ = f(y ) =, if Y = Y Then, we cn compute H(Y ). This definition for H(Y ) will be used from now nd so on solving this problem. So, we do: Now we compute λ: H(Y ) = H(Y, f(y )) = H(Y, θ) (9) = H(θ) + H(Y θ) (93) = H(λ) + P [θ = ]H(Y θ = ) + P [θ = ]H(Y θ = ) (94) = H(λ) + λh(y ) + ( λ)h(y ) (95) λ = dx (96) = (97) This mens tht Y dopts uniform distribution with probbility λ =. Now we use H(Y ) = H(λ) + λh(y ) + ( λ)h(y ) to obtin H(Y ). From tbles we know tht differentil entropy of tringle distribution is given by b +ln ( ). Where b defines the bse of the tringle. 0
11 = ( ) ln ( ) () ln () + ( ) ln ( ) + ()( + ln ()) ln() (0) Hence the mutul informtion I(X; Y ) is, I(X; Y ) = H(Y ) H(Y X) (98) = H(Y ) H(Z) (99) = H(λ) + λ Uniform {}}{ tringulr {}}{ H(Y ) +( λ) H(Y ) H(Z) (00) = λ ln (λ) ( λ) ln ( λ) + λ ln ( ) + ( λ)( + ln ()) H(Z) (0) = () ln() (03) For > we hve: f Y (y) =, (y + (+) ), ( y + (+) ), (+) y ( ) ( ) y ( ) ( ) y (+)
12 Here, we proceed the sme wy s before. Then the first step is to compute λ: So: λ =. λ = = dx (04) (05) Then H(Y ): H(Y ) = H(λ) + λ Uniform {}}{ tringulr {}}{ H(Y ) +( λ) H(Y ) (06) = λ ln (λ) ( λ) ln ( λ) + λ ln ( = ln ( ) ln ( ) + = ln ( ) + = ln () + = ln () + ( ln ( ) ln ( ) ln () + We cn compute now the mutul informtion: ) + ( λ)( + ln ()) (07) ln ( ) + ) + (08) (09) (0) () I(X; Y ) = H(Y ) H(Y X) () = H(Y ) H(Z) (3) = ln () + ln() (4) = (5)
13 For = we hve: f Y (y) = { y +, y 0 y +, 0 y Here, H(Y ) = + ln () = is esy to compute since we just hve tringulr distribution. Thus we cn compute now the mutul informtion: I(X; Y ) = H(Y ) H(Y X) (6) = H(Y ) H(Z) (7) = ln() (8) = (9) So, we cn now define the mutul informtion s follows: I(X; Y ) = ln (), 0 < <, =, > 3
14 I(X; Y ) = / ln if / if 0. As expected, I(X; Y ) s 0 nd I(X; Y ) 0 s. (b) As usul with dditive noise, we cn express I(X; Y ) in terms of h(y ) nd h(z) : (b) For = find thei(x; cpcity Y ) = of h(y the ) chnnel h(y X) when = h(y the ) input h(z) X. is pek-limited; tht is, the rnge of X is limited to / x /. Wht probbility distribution onsince X mximizes both X nd the mutul Z re informtion limited to the I (X; intervl Y)? [ /, +/], their sum Y is We limited hve tht to the for intervl =, h(z) [, = 0, +] hence:. The differentil entropy of Y is t most tht of rndom vrible uniformly distributed on tht intervl; tht is, h(y ). This mximum entropy cn be chieved I(X; ify the ) = input h(y ) X tkes on its extreme vlues (0) x = ± ech with probbility /. In this cse, I(X; Y ) = h(y ) h(z) = 0 =. By Decoding definition for Y = this X + chnnel Z, the distribution is quite simple: of Y is given by convolution between the pdfs of X nd Z. Both distributions re limited to { the intervl [, ], so the pdf of Y will be bounded by the intervl [, ]. In order to mximize / the entropy, if y < we 0 need vrible X ssuming vlues x = ± with probbilities 0.5. This ˆX wy = h(y +/ ) =. if y 0. (c) This (Optionl coding scheme from textbook) trnsmits one Findbit theper cpcity chnnel ofuse thewith chnnel zerofor error llprobbility. vlues of, (Only ginssuming received tht vluethe y = rnge 0 is mbiguous, of X is limited ndtothis / occurs x with /. probbility 0.) (c) When is of the form /m for m =, 3,..., we cn chieve the mximum possible vlue I(X; Y ) = log m when X is uniformly distributed over the discrete points {, +/(m ),..., + /(m ), +}. In this cse Y hs uniform probbility density on the intervl [ /(m ), ++/(m )]. Other vlues of re left s n exercise. 4
Tutorial 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 informationContinuous 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 informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
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 informationJoint distribution. Joint distribution. Marginal distributions. Joint distribution
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
More information8 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 information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
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 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 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 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 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 informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationSOLUTIONS 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 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 informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
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 information4.1. Probability Density Functions
STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationCS 109 Lecture 11 April 20th, 2016
CS 09 Lecture April 0th, 06 Four Prototypicl Trjectories Review The Norml Distribution is Norml Rndom Vrible: ~ Nµ, σ Probbility Density Function PDF: f x e σ π E[ ] µ Vr σ x µ / σ Also clled Gussin Note:
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 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 informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
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 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 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 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 informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
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 information15. Quantisation Noise and Nonuniform Quantisation
5. Quntistion Noise nd Nonuniform Quntistion In PCM, n nlogue signl is smpled, quntised, nd coded into sequence of digits. Once we hve quntised the smpled signls, the exct vlues of the smpled signls cn
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
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 informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationSolution to HW 4, Ma 1c Prac 2016
Solution to HW 4 M c Prc 6 Remrk: every function ppering in this homework set is sufficiently nice t lest C following the jrgon from the textbook we cn pply ll kinds of theorems from the textbook without
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 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 informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationTests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More information13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes
Integrtion by Prts 13.4 Introduction Integrtion by Prts is technique for integrting products of functions. In this Section you will lern to recognise when it is pproprite to use the technique nd hve the
More informationNormal 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 informationUNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction
Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
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 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 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 informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationThe 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 informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
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 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 informationMath 61CM - Solutions to homework 9
Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationPi evaluation. Monte Carlo integration
Pi evlution y 1 1 x Computtionl Physics 2018-19 (Phys Dep IST, Lisbon) Fernndo Bro (311) Monte Crlo integrtion we wnt to evlute the following integrl: F = f (x) dx remember tht the expecttion vlue of the
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
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 informationWe 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 informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
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 informationSection 11.5 Estimation of difference of two proportions
ection.5 Estimtion of difference of two proportions As seen in estimtion of difference of two mens for nonnorml popultion bsed on lrge smple sizes, one cn use CLT in the pproximtion of the distribution
More informationCHM Physical Chemistry I Chapter 1 - Supplementary Material
CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion
More informationFinal Exam - Review MATH Spring 2017
Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationNon-Linear & Logistic Regression
Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationBig 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 informationReview of Probability Distributions. CS1538: Introduction to Simulations
Review of Proility Distriutions CS1538: Introduction to Simultions Some Well-Known Proility Distriutions Bernoulli Binomil Geometric Negtive Binomil Poisson Uniform Exponentil Gmm Erlng Gussin/Norml Relevnce
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 informationSection 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 informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationChapter 28. Fourier Series An Eigenvalue Problem.
Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why
More informationProblem Set 3 Solutions
Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
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 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 informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
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 informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
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 informationLecture 12: Numerical Quadrature
Lecture 12: Numericl Qudrture J.K. Ryn@tudelft.nl WI3097TU Delft Institute of Applied Mthemtics Delft University of Technology 5 December 2012 () Numericl Qudrture 5 December 2012 1 / 46 Outline 1 Review
More informationPresentation Problems 5
Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
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 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 informationOverview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
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 informationMath 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
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 informationImproper 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 informationCAAM 453 NUMERICAL ANALYSIS I Examination There are four questions, plus a bonus. Do not look at them until you begin the exam.
Exmintion 1 Posted 23 October 2002. Due no lter thn 5pm on Mondy, 28 October 2002. Instructions: 1. Time limit: 3 uninterrupted hours. 2. There re four questions, plus bonus. Do not look t them until you
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationBinary Rate Distortion With Side Information: The Asymmetric Correlation Channel Case
Binry Rte Dtortion With Side Informtion: The Asymmetric Correltion Chnnel Cse Andrei Sechele, Smuel Cheng, Adrin Muntenu, nd Nikos Deliginn Deprtment of Electronics nd Informtics, Vrije Universiteit Brussel,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Deprtment 8.044 Sttisticl Physics I Spring Term 03 Problem : Doping Semiconductor Solutions to Problem Set # ) Mentlly integrte the function p(x) given in
More informationMATH 115 FINAL EXAM. April 25, 2005
MATH 115 FINAL EXAM April 25, 2005 NAME: Solution Key INSTRUCTOR: SECTION NO: 1. Do not open this exm until you re told to begin. 2. This exm hs 9 pges including this cover. There re 9 questions. 3. Do
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More information