Formula of Total Probability, Bayes Rule, and Applications

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Formula of Total Probability, Bayes Rule, and Applications"

Transcription

1 1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest. In fact, ths par of events { A, } space, herenafter denoted by S. A s a specal case of a partton of the sample 1. Partton of Sample Space and Formula of Total Probablty. Defnton of Partton. A collecton of events { S S S } 1, 2,, n of a certan sample space (or populaton) S s called a partton f () S1, S2,, Sn are mutually exclusve events; () S1 S2 Sn = S. Illustratve Example. In the nneteenth century G. Mendel conducted a famous experment that led to the frst announcement of elementary genetc prncples. He bred hybrd strans of peas and smultaneously observed the color (green or yellow) and smoothness (round or wrnkled) of the offsprng peas. If S denotes the set of all peas nvolved n the pea-breedng experment, and S 1 denotes the subpopulaton of round and green peas; S 2 denotes the subpopulaton of round and yellow peas; S 3 denotes the subpopulaton of wrnkled and green peas; S denotes the subpopulaton of wrnkled and yellow peas; 4 then {,,, } S S S S represents a partton of S Formula of Total Probablty. 1, 2,, n consttutes a partton of the sample space S. Assume that for every, 1 n, Assume that the set of events { S S S } Then for any event A, we have P( S ) > 0. (1) P( A) = P( S ) P( A S ) n. = 1

2 2 Proof. It follows from the multplcaton law of probablty that for every, 1 n, ( ) ( ) ( ) P S P A S = P AS. On the other hand, snce the events S 1, S 2,, Sn are mutually exclusve, we have that the events AS1, AS2, ASn are also mutually exclusve. In addton note that (2) AS1 AS2 ASn = A. It would be nstructve f the student tres to verfy (2) by use of a Venn dagram. Then an applcaton of the addton law of probablty to (2) gves (1). Illustratve Example. A dagnostc test for a certan dsease s known to be 95% accurate,.e., f a person has the dsease, the test wll detect t wth probablty Also, f the person does not have the dsease, the test wll report that they do not have t wth the same probablty In addton, t s known from prevous data that only 1% of the populaton has ths partcular dsease. What s the probablty that a partcular person chosen at random wll be tested postve? Soluton. Let T denote the event that a person s tested postve; T denote the event that a person s tested negatvely; D denote the event that a person has the dsease. Then t follows from the above stated condtons that we have that P T In partcular, snce P( D ) = 0.01, P( D ) = 0.99, ( D) = 0.95, and P( T D) ( ) ( ) 0.95 = P T D P T D = 1, ( D) P T = Now we apply the formula of total probablty from (1) wth A = T, n = 2, S1 = D, and S2 = D, to obtan

3 3 ( ) = ( ) ( ) ( ) ( ) P T P T D P D PT D P D = (0.95) (0.01) (0.05) (0.99) Bayes Rule. Ths mportant rule enables one to compute a condtonal probablty when the orgnal condton now becomes the event of nterest. Assume that the set { S1, S2,, Sn } consttutes a partton of the sample space S. Assume that for each, 1 n, ( ) 0 P S >. Fx any event A. Then for any gven j, 1 j n, (3) P( Sj A) = n = 1 ( j) P( A S j) P S P S ( ) P( A S ) The key thng to note about Bayes theorem s that the nformaton that wll be gven n a problem wll be the condtonal probabltes P( A S ),1 n, that appear on the rght-hand sde of the equaton, whereas what s sought s one of the P S A, where the events S j and A are reversed from the gven nformaton. I.e., gven that A occurred, what s the probablty t happened through S. condtonal probabltes, ( j ) j Proof of Bayes theorem. Note that by the multplcaton law of probablty, the numerator of the fracton on the rght-hand sde of (3) can be rewrtten as ( j) ( j) ( j) P S P A S = P AS. At the same tme, by the formula of total probablty (formula (1) above), the denomnator of the fracton on the rght-hand sde of (3) s equal to n = 1 ( ) ( ) ( ) P S P A S = P A. Hence, the rght-hand sde of (3) s equal to ( j ) P AS = P S P( A) by the defnton of condtonal probablty. ( j A).

4 4 3. Examples. Illustratve Example 1. It s qute common that dfferent llnesses produce smlar or even dentcal symptoms. Suppose that any one of the llnesses X, Y, or Z lead to the same set of symptoms, hereafter denoted as U. For smplcty assume that the llnesses X, Y, and Z are mutually exclusve and that there are no other llnesses leadng to the same set of symptoms. Suppose the probabltes of contractng these three llnesses are: P X =, PY ( ) = 0.01, PZ ( ) = 0.02, ( ) 0.03 and that the chances of developng the set of symptoms U, gven a specfc llness are: PU ( X ) = 0.85, PU ( Y ) = 0.92 PU ( Z ) = If a sck person develops the set of symptoms U, what are the chances he or she has llness X? Soluton. Frst note that the set of events X, Y, and Z together do not represent a partton. Therefore, defne H to be the event of not sufferng from any of X, Y, or Z,.e., the complement of the unon of X, Y, and Z, Then we have H = X Y Z. ( ) = 1 ( ) ( ) ( ) P H P X PY P Z = = However, PU ( H ) = 0. Applyng Bayes rule yelds that the condtonal probablty that gven the symptoms, P X U, s U that a person ndeed has the llness X, vz., ( ) PU ( X) P( X) ( ) = PU ( X) P( X) PU ( Y) PY ( ) PU ( Z) P( Z) PU ( H) P( H) P X U = (0.85) (0.03) (0.85) (0.03) (0.92) (0.01) (0.80) (0.02) 0 = Note that the data gven at the outset of the problem above nvolved the condtonal probabltes of U gven X, U gven Y, U gven Z, and U gven H, but what was sought was the condtonal probablty of X gven U, whch

5 5 nvolved the reverse of the condtons of the gven data n the problem. Ths s the prototypcal stuaton for the applcaton of Bayes theorem. Illustratve Example 2. In ths example we consder a stuaton somewhat lke the earler example above on pp. 2 and 3 of ths nsert followng the formula of total probablty. Suppose we are concerned wth medcally testng for leukema. Let T denote the event that the test s postve, suggestng the person has leukema; T denote the event that the test s negatve, suggestng the person does not have leukema; L denote the event that the person tested has leukema; L denote the event that the person tested does not have leukema. It s the case that the medcal test for leukema s not perfectly accurate. Most of the tme, f one has leukema, the test wll be postve. Past records ndcate that P( T L) 0.98 s usually negatve. Agan, t s known that PT ( L) =. Smlarly, f one does not have leukema, the test = All ths notwthstandng, there are people who sometmes test postvely, but do not, n fact, have the dsease; and some who test negatvely, but do ndeed have the dsease. If we also know that PL ( ) = , fnd: and (a) the probablty P( L T ) (b) the probablty P( L T ) of a false postve test; of a false negatve test. Soluton. (a) Note agan that n ths problem we are gven the condtonal probabltes of T gven L and T gven L, but are asked to fnd the condtonal probabltes that have the T, T events and the L, L events reversed. Hence we employ Bayes rule. Ths yelds Snce ( ) P L T = P( T L ) P( L) ( ) ( ) ( ) ( ) P T L P L P T L P L ( ) PL ( ) P L = 1 = ,.

6 6 and ( ) ( ) P T L = 1 P T L = 0.01, we obtan (0.01) ( ) P( L T ) = (0.01) ( ) (0.98) ( ) In partcular, ths mples that ( ) P( L T ) P L T = (b) Try to compute P( L T ) n an analogous manner as an exercse. (Answer: )

II. PROBABILITY OF AN EVENT

II. PROBABILITY OF AN EVENT II. PROBABILITY OF AN EVENT As ndcated above, probablty s a quantfcaton, or a mathematcal model, of a random experment. Ths quantfcaton s a measure of the lkelhood that a gven event wll occur when the

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

More information

Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

More information

Solutions to the exam in SF2862, June 2009

Solutions to the exam in SF2862, June 2009 Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

A Note on the Decomposition of a Random Sample Size

A Note on the Decomposition of a Random Sample Size A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton

More information

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2 EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei

b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei Mathematcal Propertes of the Least Squares Regresson The least squares regresson lne obeys certan mathematcal propertes whch are useful to know n practce. The followng propertes can be establshed algebracally:

More information

Solution of Algebraic and Transcendental Equations

Solution of Algebraic and Transcendental Equations CHAPTER Soluton of Algerac and Transcendental Equatons. INTRODUCTION One of the most common prolem encountered n engneerng analyss s that gven a functon f (, fnd the values of for whch f ( = 0. The soluton

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Lesson 2 Chapter Two Three Phase Uncontrolled Rectifier

Lesson 2 Chapter Two Three Phase Uncontrolled Rectifier Lesson 2 Chapter Two Three Phase Uncontrolled Rectfer. Operatng prncple of three phase half wave uncontrolled rectfer The half wave uncontrolled converter s the smplest of all three phase rectfer topologes.

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

New bounds in Balog-Szemerédi-Gowers theorem

New bounds in Balog-Szemerédi-Gowers theorem New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

More information

Lecture 2: Absorbing states in Markov chains. Mean time to absorption. Wright-Fisher Model. Moran Model.

Lecture 2: Absorbing states in Markov chains. Mean time to absorption. Wright-Fisher Model. Moran Model. Lecture 2: Absorbng states n Markov chans. Mean tme to absorpton. Wrght-Fsher Model. Moran Model. Antonna Mtrofanova, NYU, department of Computer Scence December 8, 2007 Hgher Order Transton Probabltes

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

Solution : (a) FALSE. Let C be a binary one-error correcting code of length 9. Then it follows from the Sphere packing bound that.

Solution : (a) FALSE. Let C be a binary one-error correcting code of length 9. Then it follows from the Sphere packing bound that. MATH 29T Exam : Part I Solutons. TRUE/FALSE? Prove your answer! (a) (5 pts) There exsts a bnary one-error correctng code of length 9 wth 52 codewords. (b) (5 pts) There exsts a ternary one-error correctng

More information

5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)

5. Simultaneous eigenstates: Consider two operators that commute:  η = a η (13.29) 5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let  satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB  η = a η (13.29) ˆB [  η ] = ˆB [a η ] = a

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

Moment of a force about a point and about an axis

Moment of a force about a point and about an axis 3. STATICS O RIGID BODIES In the precedng chapter t was assumed that each of the bodes consdered could be treated as a sngle partcle. Such a vew, however, s not always possble, and a body, n general, should

More information

Chapter 3 Group Theory p. 1 - Remark: This is only a brief summary of most important results of groups theory with respect

Chapter 3 Group Theory p. 1 - Remark: This is only a brief summary of most important results of groups theory with respect Chapter 3 Group Theory p. - 3. Compact Course: Groups Theory emark: Ths s only a bref summary of most mportant results of groups theory wth respect to the applcatons dscussed n the followng chapters. For

More information

PLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph

PLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph PLANAR GRAPHS Basc defntons Isomorphc graphs Two graphs G(V,E) and G2(V2,E2) are somorphc f there s a one-to-one correspondence F of ther vertces such that the followng holds: - u,v V, uv E, => F(u)F(v)

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1

IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1 Nov Sad J. Math. Vol. 36, No. 2, 2006, 0-09 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

Today s class. Chapter 13. Sources of uncertainty. Decision making with uncertainty

Today s class. Chapter 13. Sources of uncertainty. Decision making with uncertainty Today s class Probablty theory Bayesan nference From the ont dstrbuton Usng ndependence/factorng From sources of evdence Chapter 13 1 2 Sources of uncertanty Uncertan nputs Mssng data Nosy data Uncertan

More information

U.C. Berkeley CS270: Algorithms Lecture 4 Professor Vazirani and Professor Rao Jan 27,2011 Lecturer: Umesh Vazirani Last revised February 10, 2012

U.C. Berkeley CS270: Algorithms Lecture 4 Professor Vazirani and Professor Rao Jan 27,2011 Lecturer: Umesh Vazirani Last revised February 10, 2012 U.C. Berkeley CS270: Algorthms Lecture 4 Professor Vazran and Professor Rao Jan 27,2011 Lecturer: Umesh Vazran Last revsed February 10, 2012 Lecture 4 1 The multplcatve weghts update method The multplcatve

More information

Math 131: Homework 4 Solutions

Math 131: Homework 4 Solutions Math 3: Homework 4 Solutons Greg Parker, Wyatt Mackey, Chrstan Carrck October 6, 05 Problem (Munkres 3.) Let {A n } be a sequence of connected subspaces of X such that A n \ A n+ 6= ; for all n. Then S

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

Nasdaq Iceland Bond Indices 01 April 2015

Nasdaq Iceland Bond Indices 01 April 2015 Nasdaq Iceland Bond Indces 01 Aprl 2015 -Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes

More information

Solutions to First Midterm

Solutions to First Midterm rofessor Chrstano Economcs 3, Wnter 2004 Solutons to Frst Mdterm. Multple Choce. 2. (a) v. (b). (c) v. (d) v. (e). (f). (g) v. (a) The goods market s n equlbrum when total demand equals total producton,.e.

More information

QUANTUM MECHANICS, BRAS AND KETS

QUANTUM MECHANICS, BRAS AND KETS PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

H 1 : at least one is not zero

H 1 : at least one is not zero Chapter 6 More Multple Regresson Model The F-test Jont Hypothess Tests Consder the lnear regresson equaton: () y = β + βx + βx + β4x4 + e for =,,..., N The t-statstc gve a test of sgnfcance of an ndvdual

More information

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

where the coordinates are related to those in the old frame as follows.

where the coordinates are related to those in the old frame as follows. Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

9 Arithmetic and Geometric Sequence

9 Arithmetic and Geometric Sequence AAU - Busness Mathematcs I Lecture #5, Aprl 4, 010 9 Arthmetc and Geometrc Sequence Fnte sequence: 1, 5, 9, 13, 17 Fnte seres: 1 + 5 + 9 + 13 +17 Infnte sequence: 1,, 4, 8, 16,... Infnte seres: 1 + + 4

More information

Online Learning from Experts: Minimax Regret

Online Learning from Experts: Minimax Regret E0 370 tatstcal Learnng Theory Lecture 2 Nov 24, 20) Onlne Learnng from Experts: Mn Regret Lecturer: hvan garwal crbe: Nkhl Vdhan Introducton In the last three lectures we have been dscussng the onlne

More information

Halliday/Resnick/Walker 7e Chapter 25 Capacitance

Halliday/Resnick/Walker 7e Chapter 25 Capacitance HRW 7e hapter 5 Page o 0 Hallday/Resnck/Walker 7e hapter 5 apactance. (a) The capactance o the system s q 70 p 35. pf. 0 (b) The capactance s ndependent o q; t s stll 3.5 pf. (c) The potental derence becomes

More information

Lossless Data Compression

Lossless Data Compression Lossless Data Compresson Lecture : Unquely Decodable and Instantaneous Codes Sam Rowes September 5, 005 Let s focus on the lossless data compresson problem for now, and not worry about nosy channel codng

More information

A random variable is a variable whose value depends on the outcome of a random event/experiment.

A random variable is a variable whose value depends on the outcome of a random event/experiment. Random varables and Probablty dstrbutons A random varable s a varable whose value depends on the outcome of a random event/experment. For example, the score on the roll of a de, the heght of a randomly

More information

9.1 The Cumulative Sum Control Chart

9.1 The Cumulative Sum Control Chart Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

More information

VLSI Technology Dr. Nandita Dasgupta Department of Electrical Engineering Indian Institute of Technology, Madras

VLSI Technology Dr. Nandita Dasgupta Department of Electrical Engineering Indian Institute of Technology, Madras VLI Technology Dr. Nandta Dasgupta Department of Electrcal Engneerng Indan Insttute of Technology, Madras Lecture - 11 Oxdaton I netcs of Oxdaton o, the unt process step that we are gong to dscuss today

More information

Lecture 3. 1 Largest singular value The Behavior of Algorithms in Practice 2/14/2

Lecture 3. 1 Largest singular value The Behavior of Algorithms in Practice 2/14/2 18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

The eigenvalue derivatives of linear damped systems

The eigenvalue derivatives of linear damped systems Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: yjsun@su.edu.tw

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

Math 31 Lesson Plan. Day 27: Fundamental Theorem of Finite Abelian Groups. Elizabeth Gillaspy. November 11, 2011

Math 31 Lesson Plan. Day 27: Fundamental Theorem of Finite Abelian Groups. Elizabeth Gillaspy. November 11, 2011 Math 31 Lesson Plan Day 27: Fundamental Theorem of Fnte Abelan Groups Elzabeth Gllaspy November 11, 2011 Supples needed: Colored chal Quzzes Homewor 4 envelopes: evals, HW, presentaton rubrcs, * probs

More information

Gibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments)

Gibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments) Gbbs Free Energy and Chemcal Equlbrum (or how to predct chemcal reactons wthout dong experments) OCN 623 Chemcal Oceanography Readng: Frst half of Chapter 3, Snoeynk and Jenkns (1980) Introducton We want

More information

Introduction: Analysis of Electronic Circuits

Introduction: Analysis of Electronic Circuits /30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,

More information

Question 2: What is the variance and standard deviation of a dataset?

Question 2: What is the variance and standard deviation of a dataset? Queston 2: What s the varance and standard devaton of a dataset? The varance of the data uses all of the data to compute a measure of the spread n the data. The varance may be computed for a sample of

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Section B9: Zener Diodes

Section B9: Zener Diodes Secton B9: Zener Dodes When we frst talked about practcal dodes, t was mentoned that a parameter assocated wth the dode n the reverse bas regon was the breakdown voltage, BR, also known as the peak-nverse

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

The example below solves a system in the unknowns α and β:

The example below solves a system in the unknowns α and β: The Fnd Functon The functon Fnd returns a soluton to a system of equatons gven by a solve block. You can use Fnd to solve a lnear system, as wth lsolve, or to solve nonlnear systems. The example below

More information

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1 (4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at

More information

2. Linear Algebraic Equations

2. Linear Algebraic Equations 2. Lnear Algebrac Equatons Many physcal systems yeld smultaneous algebrac equatons when mathematcal functons are requred to satsfy several condtons smultaneously. Each condton results n an equaton that

More information

On some special nonlevel annuities and yield rates for annuities

On some special nonlevel annuities and yield rates for annuities On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes

More information

CHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY. Themes of the chapter. Nominal rigidities, expectational errors and employment fluctuations

CHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY. Themes of the chapter. Nominal rigidities, expectational errors and employment fluctuations CHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY Themes of the chapter Nomnal rgdtes, expectatonal errors and employment fluctuatons The short-run trade-off between nflaton and unemployment The

More information

Logistic Regression. Steve Kroon

Logistic Regression. Steve Kroon Logstc Regresson Steve Kroon Course notes sectons: 24.3-24.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro

More information

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

More information

Comment on Rotten Kids, Purity, and Perfection

Comment on Rotten Kids, Purity, and Perfection Comment Comment on Rotten Kds, Purty, and Perfecton Perre-André Chappor Unversty of Chcago Iván Wernng Unversty of Chcago and Unversdad Torcuato d Tella After readng Cornes and Slva (999), one gets the

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

More information

The complex inverse trigonometric and hyperbolic functions

The complex inverse trigonometric and hyperbolic functions Physcs 116A Wnter 010 The complex nerse trgonometrc and hyperbolc functons In these notes, we examne the nerse trgonometrc and hyperbolc functons, where the arguments of these functons can be complex numbers

More information

Non-degenerate Hilbert Cubes in Random Sets

Non-degenerate Hilbert Cubes in Random Sets Journal de Théore des Nombres de Bordeaux 00 (XXXX), 000 000 Non-degenerate Hlbert Cubes n Random Sets par Csaba Sándor Résumé. Une légère modfcaton de la démonstraton du lemme des cubes de Szemeréd donne

More information

EXPLORATION 2.5A Exploring the motion diagram of a dropped object

EXPLORATION 2.5A Exploring the motion diagram of a dropped object -5 Acceleraton Let s turn now to moton that s not at constant elocty. An example s the moton of an object you release from rest from some dstance aboe the floor. EXPLORATION.5A Explorng the moton dagram

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

( ) Homework Solutions Physics 8B Spring 09 Chpt. 32 5,18,25,27,36,42,51,57,61,76

( ) Homework Solutions Physics 8B Spring 09 Chpt. 32 5,18,25,27,36,42,51,57,61,76 Homework Solutons Physcs 8B Sprng 09 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat = E. Please refer to Fgure EX32.5. We wll choose a clockwse

More information

World Economic Vulnerability Monitor (WEVUM) Trade shock analysis

World Economic Vulnerability Monitor (WEVUM) Trade shock analysis World Economc Vulnerablty Montor (WEVUM) Trade shock analyss Measurng the mpact of the global shocks on trade balances va prce and demand effects Alex Izureta and Rob Vos UN DESA 1. Non-techncal descrpton

More information

The Mathematical Derivation of Least Squares

The Mathematical Derivation of Least Squares Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits

1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits E6 Electrcal Engneerng A rcut Analyss and Power ecture : Parallel esonant rcuts. Introducton There are equvalent crcuts to the seres combnatons examned whch exst n parallel confguratons. The ssues surroundng

More information

Homework Solutions Physics 8B Spring 2012 Chpt. 32 5,18,25,27,36,42,51,57,61,76

Homework Solutions Physics 8B Spring 2012 Chpt. 32 5,18,25,27,36,42,51,57,61,76 Homework Solutons Physcs 8B Sprng 202 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat =. Please refer to Fgure EX32.5. We wll choose a clockwse

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

THE TITANIC SHIPWRECK: WHO WAS

THE TITANIC SHIPWRECK: WHO WAS THE TITANIC SHIPWRECK: WHO WAS MOST LIKELY TO SURVIVE? A STATISTICAL ANALYSIS Ths paper examnes the probablty of survvng the Ttanc shpwreck usng lmted dependent varable regresson analyss. Ths appled analyss

More information

1. Math 210 Finite Mathematics

1. Math 210 Finite Mathematics 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information