X Z Y Table 1: Possibles values for Y = XZ. 1, p

Size: px
Start display at page:

Download "X Z Y Table 1: Possibles values for Y = XZ. 1, p"

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.

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 information

Continuous Random Variables

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

More information

Lecture 3 Gaussian Probability Distribution

Lecture 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 information

Chapter 5 : Continuous Random Variables

Chapter 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 information

Joint distribution. Joint distribution. Marginal distributions. Joint distribution

Joint 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 information

8 Laplace s Method and Local Limit Theorems

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

More information

1 The Riemann Integral

1 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 information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper 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 information

1 Probability Density Functions

1 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 information

Math 426: Probability Final Exam Practice

Math 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 information

Solution for Assignment 1 : Intro to Probability and Statistics, PAC learning

Solution 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 information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties 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 information

Lecture 1. Functional series. Pointwise and uniform convergence.

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

More information

f(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

f(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 information

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

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

More information

The Regulated and Riemann Integrals

The 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 information

Math 1B, lecture 4: Error bounds for numerical methods

Math 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 information

Lecture 21: Order statistics

Lecture 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 information

4.1. Probability Density Functions

4.1. 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 information

Review of Calculus, cont d

Review 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 information

INTRODUCTION TO INTEGRATION

INTRODUCTION 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 information

CS 109 Lecture 11 April 20th, 2016

CS 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 information

Math 360: A primitive integral and elementary functions

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

More information

A 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. 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 information

MATH 144: Business Calculus Final Review

MATH 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 information

The Wave Equation I. MA 436 Kurt Bryan

The 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 information

Best Approximation. Chapter The General Case

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

More information

7 - Continuous random variables

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

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We 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 information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition 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 information

Method: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume

Method: 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 information

15. Quantisation Noise and Nonuniform Quantisation

15. 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 information

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

The 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 information

Riemann Sums and Riemann Integrals

Riemann 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 information

Math 31S. Rumbos Fall Solutions to Assignment #16

Math 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 information

Riemann Sums and Riemann Integrals

Riemann 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 information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #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 information

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

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

More information

n 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

n 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 information

Solution to HW 4, Ma 1c Prac 2016

Solution 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 information

f(a+h) f(a) x a h 0. This is the rate at which

f(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 information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 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 information

Fundamental Theorem of Calculus

Fundamental 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 information

Tests for the Ratio of Two Poisson Rates

Tests 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 information

13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes

13.4. 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 information

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

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

More information

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

UNIT 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 information

7.2 The Definite Integral

7.2 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 information

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exam 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 information

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

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

More information

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

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

More information

1.2. 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.

1.2. 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 information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: 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 information

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

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

More information

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

W. 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 information

Definite integral. Mathematics FRDIS MENDELU

Definite 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 information

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

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

More information

Chapter 0. What is the Lebesgue integral about?

Chapter 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 information

Math 61CM - Solutions to homework 9

Math 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 information

Pi evaluation. Monte Carlo integration

Pi 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 information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite 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 information

Expectation and Variance

Expectation 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 information

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

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

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher 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 information

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

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

More information

Section 11.5 Estimation of difference of two proportions

Section 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 information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

CHM 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 information

Final Exam - Review MATH Spring 2017

Final 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 information

Non-Linear & Logistic Regression

Non-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 information

Big idea in Calculus: approximation

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

More information

Review of Probability Distributions. CS1538: Introduction to Simulations

Review 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 information

Math 8 Winter 2015 Applications of Integration

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

More information

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

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

More information

New data structures to reduce data size and search time

New 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 information

Chapter 28. Fourier Series An Eigenvalue Problem.

Chapter 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 information

Problem Set 3 Solutions

Problem 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 information

Main topics for the Second Midterm

Main 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 information

Section 17.2 Line Integrals

Section 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 information

MAA 4212 Improper Integrals

MAA 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 information

Mathematics Extension 1

Mathematics 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 information

38 Riemann sums and existence of the definite integral.

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

More information

First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009

First 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 information

3.4 Numerical integration

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

More information

Lecture 12: Numerical Quadrature

Lecture 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 information

Presentation Problems 5

Presentation 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 information

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 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 information

Theoretical foundations of Gaussian quadrature

Theoretical 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 information

Overview of Calculus

Overview 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 information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space 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 information

Numerical integration

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

More information

Math 120 Answers for Homework 13

Math 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 information

Math Solutions to homework 1

Math 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 information

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

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

More information

CAAM 453 NUMERICAL ANALYSIS I Examination There are four questions, plus a bonus. Do not look at them until you begin the exam.

CAAM 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 information

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE 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 information

Binary Rate Distortion With Side Information: The Asymmetric Correlation Channel Case

Binary 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 information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #1

MASSACHUSETTS 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 information

MATH 115 FINAL EXAM. April 25, 2005

MATH 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 information

Lecture 14: Quadrature

Lecture 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