Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016

Size: px
Start display at page:

Download "Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016"

Transcription

1 CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito detail o how to solve recurreces Recall that divide ad coquer algorithms divide up a problem ito a umber of subproblems that are the smaller istaces of the same problem, solve those problems recursively, ad combie the solutios to the subproblems ito a solutio for the origial problem Whe a subproblem size is small eough, the subproblem is simply solved i a straightforward maer I past lectures we have see two examples of divide ad coquer algorithms, merge sort ad the media-of-medias approach to the selectio problem This aturally gives rise to rutime recurreces, which express the algorithm o rutimes of iput size i terms of the rutimes of smaller iputs Recurreces Stated more techically, a divide ad coquer algorithm takes a iput of size ad does some operatios all ruig i O(f( time for some f ad rus itself recursively o k istaces of size,,, k, where i < for all i To talk about what the rutime of such a algorithm is, we ca write a rutime recurrece Recurreces are fuctios defied i terms of themselves with smaller argumets, as well as oe or more base cases We ca defie a recurrece more formally as follows: Let T ( be the rutime o istaces of size If we have k recursive calls o a give step (of sizes i ad each step takes time O(f(, the we ca write the rutime as T ( c f( k T ( i for some costat c, where our base case is T (c O( Now let s try fidig recurreces for some of the divide ad coquer algorithms we have see MergeSort Cosider the basic steps for algorithm MergeSort(A, where A Split A ito A, A of size Ru MergeSort(A ad MergeSort(A 3 Merge(A, A Steps ad 3 each take time O( Thus we kow that f( give the recurrece formula above As we are splittig the work up ito two evely sized pieces we kow that k ad each i Therefore, our fial equatio is ( T ( c T We ote that we ca replace c by by, i a sese, chagig uits We ca aalyze T ( T (/, rememberig that each cost operatio actually costs c I the ed we ca just set T ( ct ( O(T ( Thus from ow o we will remove costats as much as possible We will cosider two more recurreces The first oe correspods to a divide ad coquer algorithm for fidig the miimum umber i a (usorted array A Assume that A has size which is a power of We ca make this assumptio without loss of geerality: Suppose that k < k r for some r < k ; i

2 the we ca icrease the size of A by r by addig r ew etries at the ed that are bigger tha all itegers i the array (say The, the size of A ca at most double, ad ow it is a power of, k The algorithm FidMi(A first checks if the array size is If so, it just returs the cotets of A[0] Otherwise it recursively fids a F idmi(a[0 : / ] ad b F idmi(a[/ : ] ad returs mi{a, b} The rutime recurrece is T ( T (/ for all > ad T ( (Agai, the s i this recurrece correspod to some machie depedet costat The fial recurrece we cosider is T ( T (7/0 5 T (/5 for all > 5 ad T ( for 5 This recurrece will come up whe we itroduce a algorithm i the ext lecture 3 Methods for Solvig Recurreces Oce we have a recurrece for a fuctio, we wat to determie what the asymptotic bouds are (either Θ or O There are three mai methods for solvig recurreces Recursio Tree Substitutio Method - guess rutime ad check usig iductio 3 Master Theorem 3 Recursio Tree Recursio trees are a visual way to devise a good guess for the solutio to a recurrece, which ca the be formally proved usig the substitutio method I a recursio tree, each ode represets the cost of a sigle subproblem somewhere i the set of recursive fuctio ivocatios Summig the costs withi each level of the tree gives us the cost of each level of recursio To determie the total cost we the sum the costs of all levels of the recursio Below is a example of a recursio tree for the MergeSort algorithm At the root ode we have, represetig the size of the origial istace Sice we have two recursive calls at each step i which we divide the curret size ito two equally sized pieces, each ode as two childre that are each half the size of their paret Level 0: Level : Level : k k k Total over levels i {0,, k log }: O ( log i0 i i Recursio tree for Merge Sort LaTEX credit Mauel Kirsch k Level k: ( log O i0 O( log

3 If we aalyzed the FidMi recurrece T ( T (/ with the recursio tree method we would get that T ( O(: Level 0: Level : Level : 4 Level k: k Total over levels i {0,, k log }: log i0 i log ( While recursio trees work quite well for uderstadig simpler recurreces, more complex recurreces start to get quite tricky Below is a partial recurrece tree for the recurrece T ( T (7/05T (/5 Note that at each stage the ode is divided ito ueve groups, which will result i some leaf odes beig at differet levels tha others To get the sums i each level oe eeds to carefully calculate the differet ode sizes, ad it ca get quite messy Thus for such recurreces a differet solutio method may be preferred O( Recursio tree for selectio problem from ext lecture We ca evertheless use the recursio tree method to geerate a guess for what T ( T (7/0 5 T (/5 solves to Suppose that we drop the 5 ad terms ad aalyze T ( T (7/0 T (/5 istead We might ot get the right aswer but we ca guess that it will be somethig close to correct With the recursio tree method we see that: I level 0, the algorithm performs work I level, the sum of rutimes is 7/0 /5 9/0 3

4 I level, the sum uder the ode with 7/0 is 9/0(7/0 ad uder the ode with /5 it s 9/0(/5 Altogether it s (9/0 Iductively we ca show that i level k the algorithm performs (9/0 k work Thus the total amout of work is < i0 (9/0i 0 O( Thus we ca guess that the aswer is O( for T ( T (7/0 5 T (/5 With the ext method we will prove it by iductio 3 Substitutio Method As we saw above, recurreces trees ca get quite messy whe attemptig to solve complex recurreces With the substitutio method we ca guess what the rutime is, plug it i to the recurrece ad see if it works out Give a recurrece T ( cf( k T ( i, we ca guess that the solutio to the recurrece is i { d g(0 if T ( 0 d g( if > 0 for some costats d > 0 ad 0 ad fuctio g( We are essetially guessig that T ( O(g( For our base case we must show that you ca pick some d such that T ( 0 d g( 0, assumig T ( for istace (or whatever the base case of the recurrece is Next we assume that our guess is correct for everythig smaller tha, meaig T ( d g( for all < For our iductive hypothesis we prove the guess for We must pick some d such that f( k d g( i d g(, wheever 0 i Typically the way this works is you first try to prove the iductive step startig from the iductive hypothesis, ad the from this obtai a coditio that d eeds to obey Usig this coditio you try to figure out the base case, ie what 0 should be 3 Solvig T ( T (7/0 5 T (/5 usig Substitutio Method Recall the recurrece T ( c T ( 5 T ( for > 5 ad T ( for 5 From our recursio tree we make the guess: { d0 if T ( 0 d if > 0 For our iductive hypothesis, we assume that our guess is correct for aythig smaller tha, ad we prove our guess for That is, pick some d such that for all 0, k dg( i d g( i ( d 5 d [ 9d ( d 0 6d ] d ] d [ 9d 0 6d 4

5 Replacig the i the deomiator with 0 sice we oly care about 0, we get that we are lookig for d, 0 that satisfy the base case T ( 0 d 0 ad such that ( 9d 0 6d d 0 ( d The above is true wheever > 0 ad d If we pick 0 6 for istace we get that we ca select ay d 60 Thus we cosider the base case ad figure out what T (6 is by brute-forcig the recurrece up to 6 The we select d max{60, T (6/6} The iductio is completed 3 Issues whe usig the Substitutio Method Now we will try out a example where our guess is icorrect Cosider MergeSort, which has the recurrece T ( T ( We will guess that the algorithm is liear { d0 if T ( 0 d if > 0 We try the iductive step We try to pick some d such that for all 0, k dg( i d g( i d d ( d d d d < 0, However, the above ca ever be true, ad there is o choice of d that works! Thus our guess was icorrect This time the guess was icorrect sice MergeSort takes superliear time Sometimes however the guess ca be asymptotically correct but the iductio might ot work out Cosider for istace T ( T (/, the rutime for FidMi We kow that the rutime is O( so let s try to prove it with the substitutio method Let s guess that T ( c for all 0 First we do the iductio step: We assume that T (/ c/ ad cosider T ( We wat that c/ c However this is impossible sice the LHS is c ad the RHS is c This does t mea that T ( is ot O(, but that we chose the wrog liear fuctio We could guess istead that T ( c Now for the iductio we get (c/ c which is true for all c We ca the eve leave the base case T ( 5

6 Algorithm : Mult(x, y Split x ad y ito x 0 a b ad y 0 c d z Mult(a, c z Mult(a, d z 3 Mult(b, c z 4 Mult(b, d retur z 0 0 (z z 3 z 4 Algorithm : Karatsuba(x, y Split x 0 a b ad y 0 c d z Karatsuba(a, c z Karatsuba(b, d z 3 Karatsuba(a b, c d z 4 z 3 z z retur z 0 z 4 0 z 33 Master Method The Master Method uses the Master Theorem to solve recurreces of a special form Theorem 3 (Master Theorem A time recurrece T ( f( a T (/b, where a, b > are costats solves to: Case : T ( Θ ( log a b if f( O ( log a ɛ b for some costat ɛ > 0 domiated by the leaves of the recursio tree (ituitively: the cost is Case : T ( Θ (f( log if f( Θ ( log b a, or Case 3: T ( Θ(f( if f( Ω ( log b aɛ for some costat ɛ > 0 ad for c <, af(/b cf( (ituitively: cost is domiated by root Iteger Multiplicatio Recall the iteger multiplicatio problem, where we are give two -digit itegers x ad y ad output the product of the two digits I lecture we saw two approaches to solvig this problem Let Add( be the rutime to add two( digit itegers The, the rutime of Mult ca be writte as the recurrece T ( c Add( 4T, ad Karatsuba s rutime ca be writte as the recurrece T ( c Add( 3T ( Addig two digit itegers is a O( operatio, sice for each positio we add at most three digits: the ith digit from each umber ad possibly a carry from the additios due to the i st digits We ca the rewrite our Mult recurrece as ( T ( c 4T, ad Karatsuba as T ( c 3T ( For Mult, f( c, a 4 ad b, so log b a We ca see that f( c O( ɛ for ay ɛ, which meas case of the Master Theorem applies ad T ( Θ( For Karatsuba, f( c, a 3 ad b, so log b a log 3 59 We see that f( c O( 59 ɛ (for istace for ɛ 05, which meas case of the Master Theorem applies ad T ( O( 59 6

7 Therefore, Karatsuba s algorithm is T ( Θ( 59 wier! For iteger multiplicatio, Karatsuba is the clear 7

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

How To Solve The Homewor Problem Beautifully

How To Solve The Homewor Problem Beautifully Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

Lecture 2: Karger s Min Cut Algorithm

Lecture 2: Karger s Min Cut Algorithm priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows: Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PERFORMANCE COUNCIL (IPC) INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

More information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

Project Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments

Project Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 6-12 pages of text (ca be loger with appedix) 6-12 figures (please

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

Tradigms of Astundithi and Toyota

Tradigms of Astundithi and Toyota Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives

Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives Douglas A. Lapp Multiple Represetatios for Patter Exploratio with the Graphig Calculator ad Maipulatives To teach mathematics as a coected system of cocepts, we must have a shift i emphasis from a curriculum

More information

Solving equations. Pre-test. Warm-up

Solving equations. Pre-test. Warm-up Solvig equatios 8 Pre-test Warm-up We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Domain 1: Designing a SQL Server Instance and a Database Solution

Domain 1: Designing a SQL Server Instance and a Database Solution Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

Section 8.3 : De Moivre s Theorem and Applications

Section 8.3 : De Moivre s Theorem and Applications The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8 CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive

More information

Lecture 4: Cheeger s Inequality

Lecture 4: Cheeger s Inequality Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

Simple Annuities Present Value.

Simple Annuities Present Value. Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

Notes on exponential generating functions and structures.

Notes on exponential generating functions and structures. Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a -elemet set, (2) to fid for each the

More information

5.3. Generalized Permutations and Combinations

5.3. Generalized Permutations and Combinations 53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

More information

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k. 18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: Courat-Fischer formula ad Rayleigh quotiets The

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

7.1 Finding Rational Solutions of Polynomial Equations

7.1 Finding Rational Solutions of Polynomial Equations 4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?

More information

Time Value of Money. First some technical stuff. HP10B II users

Time Value of Money. First some technical stuff. HP10B II users Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

More information

CS100: Introduction to Computer Science

CS100: Introduction to Computer Science Review: History of Computers CS100: Itroductio to Computer Sciece Maiframes Miicomputers Lecture 2: Data Storage -- Bits, their storage ad mai memory Persoal Computers & Workstatios Review: The Role of

More information

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

Integer Factorization Algorithms

Integer Factorization Algorithms Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal

More information

Chair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics

Chair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014 1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value

More information

Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com

Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com SOLVING THE OIL DELIVERY TRUCKS ROUTING PROBLEM WITH MODIFY MULTI-TRAVELING SALESMAN PROBLEM APPROACH CASE STUDY: THE SME'S OIL LOGISTIC COMPANY IN BANGKOK THAILAND Chatpu Khamyat Departmet of Idustrial

More information

Betting on Football Pools

Betting on Football Pools Bettig o Football Pools by Edward A. Beder I a pool, oe tries to guess the wiers i a set of games. For example, oe may have te matches this weeked ad oe bets o who the wiers will be. We ve put wiers i

More information

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

More information

Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Economic Surplus and Derived Demand Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information