The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection"

Transcription

1 The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity Yamate, Suita Osaka Japa Abstract I the behavior aalysis of a compay we propose a method cosiderig subjectivity of each compay. We itroduce some motives for strategy selectio by a compay ad costruct his subjective game by usig his motive distributio. It is desirable that each compay follows his Nash equilibrium strategy for his ow subjective game. We apply this method to the Courot oligopoly market model ad obtai the subjective Nash equilibrium strategy for each player. Moreover by some umerical examples we examie the ifluece of the motive distributio o the equilibrium strategy. Keywords : The Courot oligopoly model, strategy selectio, motive distributio, subjective game. 1. Itroductio The cocept of the Nash equilibrium (Nash [5]) i o-cooperative game has bee utilized broadly i the area of micro-ecoomics. The Nash equilibrium is a equilibrium which all players reach by aimig at maximizig persoal payoff respectively. Although all players should take the Nash equilibrium strategies i order to realize this Nash equilibrium, may game experimets by huma beigs report that a player does ot Joural of Iformatio & Optimizatio Scieces Vol. 27 (2006), No. 1, pp c Taru Publicatios /06 $

2 56 S. FURUYAMA AND T. NAKAI ecessarily take his Nash equilibrium strategy. These results show that a player does ot aim at oly maximizatio of his ow payoff. The it is ecessary to explai ratioally why a player takes a strategy except his Nash equilibrium strategy. Nakai [4] otices motives for the strategy selectio by a player, deotes the ucertaity for the oppoet s motives as a motive distributio ad costructs a subjective game for each player. He proposes a subjective Nash equilibrium strategy as oe solutio of the above problem, ad explais the diversity of player s actios ratioally. Furuyama ad Nakai [2] develops the cocept of subjective game i a -perso o-cooperative game. The above-metioed argumet ca be applied eve if a player is a compay. We cosider a oligopoly market. A compay i the iterdepedece situatio geerally iteds to esure his ow profit by elaboratig all sorts of strategic thought i which he predicts others actios ad uses a artifice i order to forestall oppoets i decidig his ow actio. Game theory is ow idispesable to the aalysis i the case that the oppoet s actio iflueces this side. The traditioal game theory shows the resultat equilibrium whe all players aim at maximizig their ow payoffs. But subjective thoughts of players, for example, differeces i the stadpoits of players or the methods of approaches, etc. are ot take ito cosideratio at all. Therefore, for example i the ew etry problem, the etry compay caot discrimiate betwee the case that the existig compay has the cooperative motive to aim at the syergistic effect with the etry compay, ad the case that it has the offesive motive to stop the ew etry. This paper proposes a aalysis method for the case that each compay has oe of various motives ad that the motive is ukow for other players. That is to say, a method of behavior aalysis of a compay i cosideratio of the subjective motive uder strategy selectio is show, applyig the cocept of subjective game to the Courot model (Courot [1]) i a oligopoly market. Moreover we calculate the subjective Nash equilibrium ad examie its chage by the chage of motives i some umerical examples i order to aalyze the ifluece of motive o the equilibrium. 2. Courot oligopoly model I this sectio we cosider the traditioal Courot oligopoly model ad give its Nash equilibrium solutio. We cosider compaies

3 COURNOT OLIGOPOLY MODEL 57 (players) which produce the same goods. Let q i (q i 0, i = 1,..., ) be the quatity of productio by player i ad therefore Q = i=1 q i is the aggregate supply i the market. We assume that the market price is a liear fuctio of the aggregate supply, that is, a Q if Q a P(Q) = (1) 0 if Q a where a ( 0) is the potetial demad. Let c i be the productio cost per uit quatity of player i, π i (q 1,..., q ) be the payoff of player i whe players 1,..., produce q 1,..., q respectively, that is, π i (q 1,..., q ) = {P(Q) c i }q i (a Q c i )q i if Q a = c i q i if Q a. (2) The, we calculate the Nash equilibrium i case that compaies determie their quatities of products simultaeously. Lettig (q 1,..., q ) be the Nash equilibrium poit, q i is the solutio of max q i >0 π i(q 1,..., q i 1, q i, q i+1,..., q ). The solutio is give by q i = a c i (a c j ). (3) Therefore the aggregate supply Q, market price P(Q ) ad expected payoff π i (q 1,..., q ) of player i i the equilibrium are as follows: ( ) Q = 1 a + 1 c j (4) P(Q ) = π i (q 1,..., q ) = { ( a + ) c j a c i (5) (a c j )} 2. (6) If c i = c (c < a) (i = 1,..., ), we obtai q i = a c + 1 (7)

4 58 S. FURUYAMA AND T. NAKAI Q (a c) = + 1 (8) P(Q ) = a + c + 1 (9) π i (q 1,..., q ) = ( ) a c (10) If becomes large, the aggregate supply Q icreases, the market price P(Q ) ad the expected payoff π i of compay i decreases. Namely as the umber of compaies icreases, the competitio itesifies; as a result, the price is reduced. The, the market power of each compay becomes weak ad makes the social surplus icrease. The the icrease i the umber of compaies is a importat factor which makes a oligopolistic market trasform structurally ito a more competitive market. Therefore the chage of regulatios with respect to a ew etry has a importat meaig i market ecoomy. The literatures [3, 6, 7, 8, 9] are good guides o oligopoly market models. 3. Itroductio of subjective motives i a strategy selectio I this sectio, we costruct a model which itroduces some subjective motives for a strategy selectio by each player i the Courot oligopoly model. Nakai [4], ad Furuyama ad Nakai [2] advocate subjective games which take ito cosideratio motives for a strategy selectio by a player i the field of o-cooperative games. The subjective game is costructed by motives of all players which he expects i the competitio situatio from his subjective viewpoit. Each player will play his ow subjective game based o his thought. By this method we ca explai the behavior of a player with the altruistic character who hopes the icrease of the oppoet s profit with his self-sacrifice. There is o ivestigatio with respect to the behavior aalysis of a compay by usig a subjective game. That is to say, i the traditioal micro-ecoomics a market competitio model is costructed uder objective coditios (for example, the legal framework ad the relatio betwee supply ad demad etc.) i which the oppoet s subjective thought has ot bee cosidered explicitly. But i the behavior aalysis of a compay the maagemet attitude of the compay ad the subjective aticipatio for the attitude of the oppoet compay have importat effects. For example, sice a major compay is rich i iformatio, fuds ad techology, it ca afford to have log-term prospects, as a result, it

5 COURNOT OLIGOPOLY MODEL 59 may behave by the differet thought from a smaller compay. Therefore it has a importat sigificace that we apply the subjective game to the behavior aalysis of a compay. I this paper we propose a method for the behavior aalysis of a compay (player) by usig the above-metioed subjective game. First we itroduce some motives for a behavior of a compay. Though various motives ca be cosidered, we cosider four motives: the selfish motive, the cooperative motive, the competitive motive, the offesive motive. We cosider a motive distributio o these motives which deotes a aticipatio with respect to a motive of each player by a certai player P. Usig the origial payoff fuctio ad the motive distributio by player P, we defie a subjective game for player P. It is desirable that player P follows his Nash equilibrium strategy for his ow subjective game. We shall obtai actually the subjective game for the Courot model. As motives i selectig a strategy, we cosider four motives m 1,...,, m 4, as follows: m 1 : The selfish motive (to aim at maximizatio of his ow persoal payoff). max π i. q i >0 m 2 : The cooperative motive (to aim at maximizatio of the average of all members payoff). max q i >0 1 π j. m 3 : The competitive motive (to aim at maximizatio of the differece betwee his ow payoff ad the average of the others payoff). ( max π i 1 ) q i >0 1 π j. m 4 : The offesive motive (to aim at miimizatio of the average of the others payoff). ( max 1 ) q i >0 1 π j. Now, we cosider oe uspecified player P (ayoe of all players).

6 60 S. FURUYAMA AND T. NAKAI Let θ i = θ i 1,θi 2,θi 3,θi 4 (11) be a player i s motive distributio which player P estimates, where θ i k is the probability that player P thiks that player i follows the motive m k, θ i k 0 (k = 1,..., 4), 4 θk i k=1 = 1 (i = 1,..., ). The, whe player 1,..., produce quatities q 1,..., q respectively, the expected payoff fuctio with respect to player i s motive distributio θ i cosidered by player P is give by πi P(q 1,..., q θ i ) = θ1 i π i + θ2 i 1 +θ i 4 = θ i 1 π i + θi 2 = θi 4 ( π j + θ3 i π i 1 ) 1 π j ( 1 ) 1 π j 1 ( ( π j + θ3 i π i θi 3 ) π 1 j π i π j π i ) ( θ i 1 + θi 3 + θi 3 + θi 4 1 ) θ i π i +( 2 θi 3 + ) θi 4 π 1 j = A i π i + B i π j (12) where A i = θ i 1 + θi 3 + θi 3 + θi 4 1 (13) B i = θi 2 θi 3 + θi 4 1. (14) By the equatio (2), we obtai πi P(q 1,..., q θ i ) (A i +B i )qi 2 + {(A i +B i )(a c i ) (A i +2B i )Q i }q ) i = +B i (aq i Q 2 i c j q j if Q a (A i + B i )c i q i B i c j q j if Q a (15)

7 COURNOT OLIGOPOLY MODEL 61 where Q i = q j. We call the above-metioed o-cooperative game (π1 P,..., π P ) the subjective Courot game for player P. 4. Subjective Nash equilibrium I this sectio, we shall obtai a Nash equilibrium of the subjective Courot game for player P defied i the previous sectio. The Nash equilibrium i this game is called the subjective Nash equilibrium for player P. Let q = (q 1,..., q ) be the subjective Nash equilibrium poit. Furthermore we put Q = q j, Q i = q j. The q i is the solutio of the followig maximizig problem π P i (q 1,..., q i 1, q i, q i+1,..., q θ i ) max q i ( 0). (16) From the equatio (15), the fuctio πi P is a cocave quadratic fuctio with respect to q i i regio q i a Q i ad a decreasig liear fuctio i regio q i a Q i. Furthermore the fuctio π i P is cotiuous at q i = a Q i. The the optimal solutio q i exists i the iterval [0, a Q i ]. Whe 0 q i a Q i, from π i P/ q i = 0, we obtai q i = (A i + B i )(a c i ) (A i + 2B i )Q i 2(A i + B i ) (17) if A i + B i = 0, that is, θ i 4 < 1. Substitutig Q i = Q qi i the equatio (17) ad rewritig it, we obtai q i = (A i + B i )(a c i ) (A i + 2B i )Q A i (18) if A i = 0, that is, θ i 2 < 1. Calculatig the sum Q = K j (a c j ) 1 + (2K j 1) of (18), we obtai i=1, (19)

8 62 S. FURUYAMA AND T. NAKAI where K i = 1 + B i = ( 1)θi 1 + ( 1)θi 2 + ( 1)θi 3 A i ( 1)θ1 i + 2 θ3 i +. (20) θi 4 Substitutig equatio (19) i equatio (18), q i = K i (a c i ) 2K i K j (a c j ) q i. (21) (2K j 1) Let q i be the value of the right had side of the equatio (21). I order that q i is optimal, it must satisfy the coditio 0 q i a Q i. The to be exact, q i = mi{max( q i, 0), a Q i }. (22) The the followig theorem is acquired. Theorem 1. Suppose that θ2 i = 1, θi 4 = 1 for ay i, amely, player P cosiders that o player is cooperative ad offesive with probability 1. The subjective Nash equilibrium poit for player P is give by q = (q 1,..., q ) where 0 if q i < 0 qi = q i if 0 q i a Q (23) i a Q i if a Q i < q i. The value q i is give by (21). The coditio q i a Q i is equivalet to the followig relatio: K j (a c j ) 1 + (2K j 1) a. (24) The subjective Nash equilibrium strategy of player P is q P. The equilibrium price is P(Q ) = a Q = a K j (a c j ) 1 + (2K j 1). (25)

9 COURNOT OLIGOPOLY MODEL 63 The expected payoff of player i at the subjective Nash equilibrium poit is π P i (q 1,..., q θ i ) = (a Q c i )q i 5. Numerical examples = K i (a c i ) 2 (a c i )(3K i 1) +(2K i 1) { K j (a c j ) 1 + (2K j 1) K j (a c j ) 1 + (2K j 1) } 2. (26) I this sectio we give some umerical examples showig the equilibrium productio quatity ad the equilibrium payoff for each player uder various motive distributios. I all examples we suppose that = 3, a = 1000, c i = c = 2 ad that θ2 i = 1, θi 4 = 1 (i = 1, 2, 3). Moreover we restrict our attetio to the case that the costrait (24) is satisfied, that is, where 499(K 1 + K 2 + K 3 ) K 1 + K 2 + K (27) K i = 2(3θi 1 + θi 2 + 3θi 3 ) 3(2θ i 1 + 3θi 3 + θi 4 ). (28) The i the followig figures, it may occur that the graph breaks halfway. Example 1. We cosider the case that all players have oly two motives m 1 ad m 2 ad have the same motive distributio θ i 1,θi 2,θi 3,θi 4 = 1 θ,θ, 0, 0 (i = 1, 2, 3) where θ is the probability that each player follows the cooperative motive m 2. The commo equilibrium productio quatity q i is show by the curve m 2 i Figure 1. The curve m 2 is decreasig getly. Namely if all players become more cooperative, the each player aims to maximize the social payoff (the average of all player s payoffs) ad refrais from high productio. Similarly we cosider the case that all players have the same motive distributio 1 θ, 0, θ, 0 ( 1 θ, 0, 0, θ ). The commo equilibrium productio quatity is show by the curve m 3 (m 4 ) i Figure 1. The curve m 3 is icreasig getly. This meas that if all players become more

10 64 S. FURUYAMA AND T. NAKAI competitive, the each player icreases his productio quatity, as a result, the society falls i overproductio. The curve m 4 is icreasig rapidly. Namely if all players become more offesive, the each player icreases his productio quatity explosively, as a result, the price falls heavily. Figure 1 The equilibrium productio quatity i Example 1 Example 2. We cosider the case that player II ad III have the same motive distributio 0.7, 0.1, 0.1, 0.1, that is, they are, if aythig, selfish ad that player I follows the motive distributio 1 θ, 0,θ, 0 ( 1 θ, 0, 0,θ ). The equilibrium productio quatity qi ad the equilibrium payoff πi P for each player are show i Figure 2, 3 (Figure 4, 5). From Figure 2, 3 we kow the followigs: If player I becomes more competitive, the he ca expect the icrease of his ow payoff by icreasig his productio quatity. O the other had player II ad III must decrease their productio quatities, as a result, their expected payoffs decrease. Namely player I is active ad player II ad III are passive. From Figure 4, 5, we kow the followigs: If player I becomes more offesive, the the above-metioed tedecy becomes large, but the equilibrium payoff of player I decreases rapidly after passig the peak sice the decrease of the other players payoffs reach the limit. Example 3. We suppose that player II ad III have motive distributio 0.7, 0.1, 0.1, 0.1 ad 0.1, 0.1, 0.7, 0.1 respectively, that is, player II (III) is, if aythig, selfish (competitive). Whe the motive distributio of player I is 1 θ, 0,θ, 0) ( 1 θ, 0, 0,θ ), the equilibrium productio

11 COURNOT OLIGOPOLY MODEL 65 Figure 2 The equilibrium productio quatity i the case of 1 θ, 0,θ, 0 i Example 2 Figure 3 The expected payoff i the case of 1 θ, 0,θ, 0 i Example 2 Figure 4 The equilibrium productio quatity i the case of 1 θ, 0, 0,θ i Example 2

12 66 S. FURUYAMA AND T. NAKAI Figure 5 The expected payoff i the case of 1 θ, 0, 0,θ i Example 2 quatity ad the equilibrium payoff of each player are show i Figure 6, 7 (Figure 8, 9). From Figure 6, 7 we kow the followigs: If player I becomes more competitive, the his productio quatity icreases ad the productio quatities of player II ad III decrease slightly. O the other had the equilibrium payoff of player I chages from the slight icrease to the slight decrease ad the equilibrium payoffs of player II ad III decrease. From Figure 8, 9 we kow the followigs: If player I becomes more offesive, the the above-metioed tedecy becomes large. The equilibrium payoff of player I decreases rapidly as the expected payoffs of other players decrease. Figure 6 The equilibrium productio quatity i the case of 1 θ, 0,θ, 0 i Example 3

13 COURNOT OLIGOPOLY MODEL 67 Figure 7 The expected payoff i the case of 1 θ, 0,θ, 0 i Example 3 Figure 8 The equilibrium productio quatity i the case of 1 θ, 0, 0,θ i Example 3 Figure 9 The expected payoff i the case of 1 θ, 0, 0,θ i Example 3

14 68 S. FURUYAMA AND T. NAKAI 6. Cocludig commets I this paper, we itroduced the behavior aalysis model of the compay cosiderig the subjective motive i the strategy selectio. The fudametal model is the Courot oligopoly model. By applyig the subjective game to it, we reflect the subjective thoughts of compaies with differet stadpoits ad attitudes, etc. i decisio-makig. We cosider four types of motives for the costructio of the model, ad give three umerical examples. Of course, more motives ca be cosidered actually. With the icrease of the umber of motives, the equilibrium aalysis seems to become more complicated. From umerical examples we ca kow that i the Courot oligopoly market the o-cooperative motive (competitive motive ad offesive motive) brigs a advatageous result. I the traditioal ecoomics may discussios have bee made o the assumptio of the perfect ratioality ad objective coditios of the market. But there are a few ivestigatios treatig subjectivities of players. The our model with subjective game is hoped for oe of market aalysis models pursuig subjectivities of compaies. Refereces [1] A. Courot, Researches ito the Mathematical Priciples of the Theory of Wealth, N. Baco (ed.), Macmilla, New York, [2] S. Furuyama ad T. Nakai, The costructio of subjective games by motive distributios i -perso o-cooperative game, Joural of Iformatio ad Optimizatio Scieces, Vol. 25 (3) (2004), pp [3] R. Gibbos, Game Theory for Applied Ecoomists, Priceto Uiversity Press, [4] T. Nakai, Subjective games i a o-cooperative game, Joural of Iformatio ad Optimizatio Scieces, Vol. 21 (1) (2000), pp [5] J. F. Nash, No-cooperative games, Aals of Mathematics, Vol. 54 (1951), pp [6] L. Phlips, Applied Idustrial Ecoomics, Cambridge Uiversity Press, [7] W. G. Shepherd ad J. M. Shepherd (2004), The Ecoomics of Idustrial Orgaizatio, Wavelad Press, Ic. [8] O. Shy, Idustrial Orgaizatio, The MIT Press, [9] J. Tirole, The Theory of Idustrial Orgaizatio, The MIT Press, Received March, 2005

Problem Set 1 Oligopoly, market shares and concentration indexes

Problem Set 1 Oligopoly, market shares and concentration indexes Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet

More information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

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

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

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

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

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

Building Blocks Problem Related to Harmonic Series

Building Blocks Problem Related to Harmonic Series TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

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

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

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

Alternatives To Pearson s and Spearman s Correlation Coefficients

Alternatives To Pearson s and Spearman s Correlation Coefficients Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives

More information

Confidence Intervals for One Mean with Tolerance Probability

Confidence Intervals for One Mean with Tolerance Probability Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

More information

ARITHMETIC AND GEOMETRIC PROGRESSIONS

ARITHMETIC AND GEOMETRIC PROGRESSIONS Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

More information

Amendments to employer debt Regulations

Amendments to employer debt Regulations March 2008 Pesios Legal Alert Amedmets to employer debt Regulatios The Govermet has at last issued Regulatios which will amed the law as to employer debts uder s75 Pesios Act 1995. The amedig Regulatios

More information

Riemann Sums y = f (x)

Riemann Sums y = f (x) Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

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

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

France caters to innovative companies and offers the best research tax credit in Europe

France caters to innovative companies and offers the best research tax credit in Europe 1/5 The Frech Govermet has three objectives : > improve Frace s fiscal competitiveess > cosolidate R&D activities > make Frace a attractive coutry for iovatio Tax icetives have become a key elemet of public

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

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Power Factor in Electrical Power Systems with Non-Linear Loads

Power Factor in Electrical Power Systems with Non-Linear Loads Power Factor i Electrical Power Systems with No-Liear Loads By: Gozalo Sadoval, ARTECHE / INELAP S.A. de C.V. Abstract. Traditioal methods of Power Factor Correctio typically focus o displacemet power

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract

The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract The Gompertz Makeham couplig as a Dyamic Life Table By Abraham Zaks Techio I.I.T. Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 32000, Haifa, Israel Abstract A very famous

More information

A Study for the (μ,s) n Relation for Tent Map

A Study for the (μ,s) n Relation for Tent Map Applied Mathematical Scieces, Vol. 8, 04, o. 60, 3009-305 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.04.4437 A Study for the (μ,s) Relatio for Tet Map Saba Noori Majeed Departmet of Mathematics

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

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totient, the Möbius and the Divisor Functions The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

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

9.8: THE POWER OF A TEST

9.8: THE POWER OF A TEST 9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

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

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

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

Standard Errors and Confidence Intervals

Standard Errors and Confidence Intervals Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

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

Tagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper Part-A

Tagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper Part-A Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper Part-A Uit-I. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

Introductory Explorations of the Fourier Series by

Introductory Explorations of the Fourier Series by page Itroductory Exploratios of the Fourier Series by Theresa Julia Zieliski Departmet of Chemistry, Medical Techology, ad Physics Momouth Uiversity West Log Brach, NJ 7764-898 tzielis@momouth.edu Copyright

More information

Subject CT5 Contingencies Core Technical Syllabus

Subject CT5 Contingencies Core Technical Syllabus Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value

More information

Systems Design Project: Indoor Location of Wireless Devices

Systems Design Project: Indoor Location of Wireless Devices Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

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

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

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

More information

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S, Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Hypothesis Tests Applied to Means

Hypothesis Tests Applied to Means The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

More information

G r a d e. 5 M a t h e M a t i c s. Patterns and relations

G r a d e. 5 M a t h e M a t i c s. Patterns and relations G r a d e 5 M a t h e M a t i c s Patters ad relatios Grade 5: Patters ad Relatios (Patters) (5.PR.1) Edurig Uderstadigs: Number patters ad relatioships ca be represeted usig variables. Geeral Outcome:

More information

Journal of Chemical and Pharmaceutical Research, 2015, 7(3):1184-1190. Research Article

Journal of Chemical and Pharmaceutical Research, 2015, 7(3):1184-1190. Research Article Available olie www.ocpr.com Joural of Chemical ad Pharmaceutical Research, 15, 7(3):1184-119 Research Article ISSN : 975-7384 CODEN(USA) : JCPRC5 Iformatio systems' buildig of small ad medium eterprises

More information

Review for College Algebra Final Exam

Review for College Algebra Final Exam Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 1-4. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i

More information

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively

More information

Section IV.5: Recurrence Relations from Algorithms

Section IV.5: Recurrence Relations from Algorithms Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

More information

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

More information

Confidence Intervals and Sample Size

Confidence Intervals and Sample Size 8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGraw-Hill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 7-1 Cofidece Itervals for the

More information

BASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.

BASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1. BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts

More information

Grade 7. Strand: Number Specific Learning Outcomes It is expected that students will:

Grade 7. Strand: Number Specific Learning Outcomes It is expected that students will: Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]

More information

Numerical Solution of Equations

Numerical Solution of Equations School of Mechaical Aerospace ad Civil Egieerig Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 TPFE MSc CFD- Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg,

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

Unit 20 Hypotheses Testing

Unit 20 Hypotheses Testing Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect

More information

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville Real Optios for Egieerig Systems J: Real Optios for Egieerig Systems By (MIT) Stefa Scholtes (CU) Course website: http://msl.mit.edu/cmi/ardet_2002 Stefa Scholtes Judge Istitute of Maagemet, CU Slide What

More information

Pre-Suit Collection Strategies

Pre-Suit Collection Strategies Pre-Suit Collectio Strategies Writte by Charles PT Phoeix How to Decide Whether to Pursue Collectio Calculatig the Value of Collectio As with ay busiess litigatio, all factors associated with the process

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

Using Excel to Construct Confidence Intervals

Using Excel to Construct Confidence Intervals OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

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

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

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

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat

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

represented by 4! different arrangements of boxes, divide by 4! to get ways

represented by 4! different arrangements of boxes, divide by 4! to get ways Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,

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

Section 7-3 Estimating a Population. Requirements

Section 7-3 Estimating a Population. Requirements Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio

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

B1. Fourier Analysis of Discrete Time Signals

B1. Fourier Analysis of Discrete Time Signals B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)

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

Measurable Functions

Measurable Functions Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

More information

The Nine Dots Puzzle Extended to nxnx xn Points

The Nine Dots Puzzle Extended to nxnx xn Points The Nie Dots Puzzle Exteded to xx x Poits Marco Ripà 1 ad Pablo Remirez 2 1 Ecoomics Istitutios ad Fiace, Roma Tre Uiversity, Rome, Italy Email: marcokrt1984@yahoo.it 2 Electromechaical Egieerig, UNLPam,

More information

Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition

Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition 7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

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

3. Covariance and Correlation

3. Covariance and Correlation Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

More information

Is there employment discrimination against the disabled? Melanie K Jones i. University of Wales, Swansea

Is there employment discrimination against the disabled? Melanie K Jones i. University of Wales, Swansea Is there employmet discrimiatio agaist the disabled? Melaie K Joes i Uiversity of Wales, Swasea Abstract Whilst cotrollig for uobserved productivity differeces, the gap i employmet probabilities betwee

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

Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...

Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,... 3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.

More information

ODBC. Getting Started With Sage Timberline Office ODBC

ODBC. Getting Started With Sage Timberline Office ODBC ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.

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

Concept #1. Goals for Presentation. I m going to be a mathematics teacher: Where did this stuff come from? Why didn t I know this before?

Concept #1. Goals for Presentation. I m going to be a mathematics teacher: Where did this stuff come from? Why didn t I know this before? I m goig to be a mathematics teacher: Why did t I kow this before? Steve Williams Associate Professor of Mathematics/ Coordiator of Secodary Mathematics Educatio Lock Have Uiversity of PA swillia@lhup.edu

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

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

Section 6.1. x n n! = 1 + x + x2. n=0

Section 6.1. x n n! = 1 + x + x2. n=0 Differece Equatios to Differetial Equatios Sectio 6.1 The Expoetial Fuctio At this poit we have see all the major cocepts of calculus: erivatives, itegrals, a power series. For the rest of the book we

More information

AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

More information

Capacity of Wireless Networks with Heterogeneous Traffic

Capacity of Wireless Networks with Heterogeneous Traffic Capacity of Wireless Networks with Heterogeeous Traffic Migyue Ji, Zheg Wag, Hamid R. Sadjadpour, J.J. Garcia-Lua-Aceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata

More information

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio

More information

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model 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

Fourier Series and the Wave Equation Part 2

Fourier Series and the Wave Equation Part 2 Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries

More information