3 Basic Definitions of Probability Theory

Save this PDF as:

Size: px
Start display at page:

Download "3 Basic Definitions of Probability Theory"

Transcription

1 3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio ecompasses the Classical ad Frequecy defiitios of probability Note that i a umber of places i these otes I use axiom as a syomom for assumptio. Classical probability Defiitio Classical probability (MBG): If a radom experimet (process with a ucertai outcome) ca result i mutually exclusive ad equally likely outcomes, ad if A of these outcomes has a attribute A, the the probability of A is the fractio A /. This otio of probabilty had its coceptio i the study of games of chace; i particular, fair games of chace. E.g. if oe tosses a coi there are two mutually exclusive outcomes: head or tail. Of these two outcomes, oe is associated with the attribute heads; oe is associated with the attribute tails. If the coi is fair each outcome is equally likely. I which case, Pr head A 1, 2 where 2ad A is the umber of possible outcomes associated with a head (1). Cosider some other examples: 1. The roll of a die: There are 6 equally likely outcomes. The probability of each is 1/6. 2. Draw a card from a deck: There are 52 equally likely outcomes. 3. The roll of two die: There are 36 equally likely outcomes (6x6): 6 possibilities for the first die, ad 6 for the secod. The probability of each outcome is 1/36.

2 4. Drawig (with replacemet) four balls from a ur with a equal umber of red, white, ad blue balls: There are 81 possible outcomes (3x3x3x3 3 4 ). For example, {red, white, white, blue} is a outcome which is a differet outcome from {white, white, red, blue}. The probability associated with each outcome is 1/ The toss of two cois: The four possible outcomes are H,H, H,T, T,H ad TT.The probability of each is 1/4. 6. The draw of two cards: There are 52 2 possible outcomes. Terms to ote i the defiitio of classical probability are radom,, mutually exclusive, ad equally likely. Axiom A Basic assumptio i the defiitio of classical probability is that is a fiite umber; that is, there is oly a fiite umber of possible outcomes. If there is a ifiite umber of possible outcomes, the probability of a outcome is ot defied i the classical sese. Defiitio mutually exclusive: The radom experimet result i the occurace of oly oe of the outcomes. E.g. if a coi is tossed, the result is a head or a tail, but ot both. That is, the outcomes are defied so as to be mutually exclusive. Defiitio equally likely: Each outcome of the radom experimet has a equal chace of occurig. Defiitio radom experimet (Gujarati p23): A radom experimet is a process leadig to at least two possible outcomes with ucertaity as to which will occur. Defiitio sample space: The collectio of all possible outcomes of a experimet. If I had to guess, I would say it is called sample space because it is the collectio (set) of all possible samples. As importat thig to ote is that classical probabilities ca be deduced from kowledge of the sample space ad the assumptios. Nothig has to be observed i terms of outcomes to deduce the probabilities. Frequecy Probability What if is ot fiite? I that case, the Classical defiitio is ot applicable. What if the outcomes are ot equally likely? Agai, the Classical defitio of probability is ot applicable. I such cases, how might we defie the probablity of a outcome that has attribute A.

3 Defiitio We might take a radom sample from the populatio of iterest ad idetify the proportio of the sample with attribute A. That is, calculate umber of obser i the sample that possess attribute A Relative freq of A i the sample umber of obser i the sample The assume Relative freq of A i the sample is a estimate of Pr A The foudatio of this approach is that there is some Pr A. We caot deduce it, as i Classical probability, but we ca estimate it. For example, oe tosses a coi, which might or might ot be fair, 100 times ad observes heads o 52 of the tosses. Oe s estimate of the probability of a head is.52. Frequecy probability allows to estimate probabilities whe Classical probability provides o isight. Axiomatic Approach to Probability Put simply, the axiomatic approach build up probability theory from a umber of assumptios (axioms). Axiom there is some sample space, : the collectio of all possible outcomes of a experimet For example, if the experimet is tossig a coi H,T. If the experimet is rollig a die 1,2,3,4,5,6. If the experimet is tossig two cois H,H, H,T, T,H, T,T. If the experimet is takig a sample of oe from the U.S. populatio, populatio of the U.S.. If the experimet is drawig a radom iterger, is the set of all itergers (there are a ifiite umber of these). Defiitio Evet A (MGB p15): a subset of the sample space. The set of all evets associated with a experimet is defied as the evet space Note that the above is ot truly a defitio (it does ot idetify the ecc ad suff coditios). Beig a subset of (A ) is a ecessary but ot a sufficiet coditio for beig a evet. That is, for some sample spaces, there are subset that are ot evets; evet space ca be a strict subset of sample space. If the umber of outcomes i is fiite, the all subsets of are evets.

4 I geeral, oe of the outcomes is a evet, typically deoted, ad oe of the outcomes is a evet, deoted. Thatis, is both the sample space ad a evet, a certai evet. The evet set is difficult to formally defie. Some examples of evet sets: 1. the toss of a sigle coi: H,T. The possible evets are H, T, either H or T ( ), ad either H or T ( ). 2. the toss of two cois: H,H, H,T, T,H, T,T. How may evets are there? (H,H), (H,T), (T,H), (T,T), oe of the these outcomes, oe of these outcomes, at least oe head ( (H,H), (H,T) or(t,h)), at least oe tail ((H,T), (T,H) or(t,t)). Ca you thik of others? 3. drawig 4 cards from a deck: Evets iclude all spades, sum of the 4 cards is 20 (assumig face cards have a value of zero), a sequece of itegers, a had with a 2,3,4 ad 5. There are may more evets. 4. the example o page 16 of MGB: radomly choose a lightbulb to observe ad record time it takes to bur out. x : x 0 Defie evet A k,m as A x : k x m I ecoometrics we will be cocered with the probablility of evets, ad eed oly to defie those evets we care about. For example, we might be cocered with the probability that the wier is a female or that the sample mea is 5 give that the populatio mea is 0. Typically, oe uses capital letters to represet evets. E.g. A, B, C; ora 1, A 2, A 3, etc. I will use Ą to repeset the set of all possible evets. Axiom Ą. That is, the evet that oe of the outcomes is occurs i a evet. Note that Pr 1. Axiom if A Ą the Ā Ą where Ā is the complimet of A. That is, if A is a evet, the ot A is a evet Axiom if A 1 ad A 2 AtheA 1 A 2 Ą. That is two evets together is a evet. These three assumptios imply the followig: 1. Ą For example, this implicatio follows from the first two Axioms. 2. if A 1,A 2,...A Ą the i 1 A i ad i 1 A i Ą

5 Ay collectio of evets that fulfills the above three assumptios/axioms is called a Boolea algebra Defiitio The axiomatic defitio of probability: With these defiitios ad assumptios i mid, a probability fuctio Pr. maps evets i Ą oto the 0,1 iteval. That is, there exists some fuctio that idetifies the probabilty associated with ay evet i Ą. It fufills the followig axioms: Pr A 0 A Ą Pr 1 If A 1,A 2,...,A is a sequece of mutually exclusive evets (A i A j i,j i j) ad if i 1 A i Ą the Pr i 1 A i i 1 Pr A i. From this defiitio with its three axioms, it is possible to deduce a buch of properties that oe would expect a probability fuctio to have (MGB 24). Icludig Pr 0, ad Pr Ā 1 Pr A If A ad B Ą ad A B the Pr A Pr B If A 1,A 2,...A Ą the Pr A 1 A 2... A Pr A 1 Pr A 2... Pr A. This is called Boole s iequality. If A ad B Ą the Pr A Pr A B Pr A B. That is, Pr A Pr AB Pr AB If A ad B Ą the Pr A B Pr A Pr A B. Thatis,Pr A B Pr A Pr AB Put simply, the axiomatic defiitio builds up the otio of a probability fuctio from a umber of assumptios/axioms. The properties of that probablity fuctio follow from the assumptios. Note that if the umber of outcomes are fiite ad equally likely the oe has the Classical world of probability. Also ote that the Frequecy defitio assumes the existece of the probability fuctio Pr A. The axiomatic approach subsumes the Classical ad Frequecy approaches. Probability of evet A coditioal o evet B occurig Let Pr A B Probability of evet A coditioal o evet B occurig, assumig Pr B 0. Otherwise Pr A B is ot defied.

6 Cosider the followig: x,y :0 x 100, 0 y 100. Now defie two sets: A x,y :20 x 50, 40 y 60 ad B x,y :0 x 40, 50 y 100. Note that these two sets defie evets. Note that i this example the two sets partially itersect. Cosider other examples where A B or A B. Draw some pictures. Defiitio of coditioal probability (1) Pr A B Pr AB Pr B if Pr B 0. ReadPr AB Pr A B Pr A ad B. So (2) Pr B A Pr AB Pr A if Pr A 0. Oe ca rearrage (1) ad (2) to obtai (1a) ad (2a) ad Combiig (1a) ad (2a), oe obtais (3) Pr AB Pr A B Pr B Pr BA Pr B A Pr A Pr AB Pr A B Pr B Pr B A Pr A Pr BA What is the ituitio behid this beig the defiitio of coditioal probability? I coditioal probability, the sample space is effectively reduced to B; that is, what is the probability that A will happe give that oes lives i a world where B prevails. So, Pr A B is the probability that both A ad B occur, Pr AB, as a proportio of the probability of B, Pr B. Note the followig, which followig from (2) ad (3): Pr B A Pr AB Pr A Pr A B Pr B Pr A This says that if oe kows, Pr A B, Pr B, adpr A, oe ca determie Pr B A. What is this result called? Baye s theorem? Whe are evets A ad B idepedet?

7 Start by cosiderig a case where Pr A 0adPr B 0, but A B, sopr AB 0. If these assumptios hold, Pr A B Pr B A 0; that is, the two evets are mutually exclusive. Draw a Ve diagram. I this case, are A ad B idepedet? NO. Oe precludes the other. Whe do we say A ad B are idepedet? Defiitio A ad B are idepedet iff ay of the followig are true: Pr AB Pr A B Pr A Pr B Pr A B Pr A if Pr B 0 Pr B A Pr B if Pr A 0 Note that these are three equivalet statemets: each implies the other two. Ca you show this? The followig ca be deduced: Pr A 0, Pr B 0adA B A ad B are ot idepedet Pr A 0, Pr B 0adA ad B are idepedet A B Idepedece of evets is more complicated tha idepedece of 2 evets. Some studet examples of idepedece ad depedece Example 1 Suppose a survey classified the populatio as male or female, ad as favorig or opposig the death pealty. Suppose, the proportios i each category were Death ot Death Male There are four evets: male, female, favor death pealty, Female do t favor death pealty. I this case, the probability of a idividual favorig the death pealty, coditioal o beig male is Pr D M Pr D,M Pr M

8 ad Pr D , so the the probability of favorig the death pealty equals the probability of favorig the death pealty coditioal o oe beig male, so the two evets (favorig death ad beig male) are idepedet. However if the proportios were Death ot Death Male Female Pr D M Pr D,M Pr M ad Pr D , so the the probability of favorig the death pealty does ot equal the probability of favorig the death pealty coditioal o oe beig male, so the two evets (favorig death ad beig male) are depedet. Males are more likely to prefer the death pealty.

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

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

More information

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

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

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

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

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

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

More information

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

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

MARTINGALES AND A BASIC APPLICATION

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

More information

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

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the

More information

Infinite Sequences and Series

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

More information

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

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

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

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

Overview of some probability distributions.

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

More information

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

A probabilistic proof of a binomial identity

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

More information

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

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

More information

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

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

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

Hypothesis testing. Null and alternative hypotheses

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

More information

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

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

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

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

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

More information

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

More information

Soving Recurrence Relations

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

More information

Betting on Football Pools

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

More information

Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Span, linear independence, bases, and dimension Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

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

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

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 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

A Mathematical Perspective on Gambling

A Mathematical Perspective on Gambling A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal

More information

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

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

More information

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

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

More information

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

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

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

More information

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

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

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

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

More information

Math C067 Sampling Distributions

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

More information

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

4.3. The Integral and Comparison Tests

4.3. The Integral and Comparison Tests 4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece

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

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

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

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

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

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

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

Present Values, Investment Returns and Discount Rates

Present Values, Investment Returns and Discount Rates Preset Values, Ivestmet Returs ad Discout Rates Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC dmidli@cdiadvisors.com May 2, 203 Copyright 20, CDI Advisors LLC The cocept of preset value lies

More information

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

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

More information

5 Boolean Decision Trees (February 11)

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

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Maximum Likelihood Estimators.

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

More information

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

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

More information

A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING

A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING MATTHEW ACTIPES Abstract. This paper begis by defiig a probability space ad establishig probability fuctios i this space over discrete radom variables.

More information

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

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

More information

INFINITE SERIES KEITH CONRAD

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

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

Lesson 17 Pearson s Correlation Coefficient

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

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

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

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

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

More information

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

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

More information

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book) MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

More information

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

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

More information

Institute of Actuaries of India Subject CT1 Financial Mathematics

Institute of Actuaries of India Subject CT1 Financial Mathematics Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i

More information

Normal Distribution.

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

More information

Elementary Theory of Russian Roulette

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

More information

Uncertainty Chapter 13. Mausam (Based on slides by UW-AI faculty)

Uncertainty Chapter 13. Mausam (Based on slides by UW-AI faculty) Ucertait Chapter 3 Mausam Based o slides b UW-AI facult Kowledge Represetatio KR Laguage Otological Commitmet Epistemological Commitmet ropositioal Logic facts true false ukow First Order Logic facts objects

More information

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

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

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

One-sample test of proportions

One-sample test of proportions Oe-sample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:

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

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio

More information

Research Method (I) --Knowledge on Sampling (Simple Random Sampling)

Research Method (I) --Knowledge on Sampling (Simple Random Sampling) Research Method (I) --Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact

More information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

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

Central Limit Theorem and Its Applications to Baseball

Central Limit Theorem and Its Applications to Baseball Cetral Limit Theorem ad Its Applicatios to Baseball by Nicole Aderso A project submitted to the Departmet of Mathematical Scieces i coformity with the requiremets for Math 4301 (Hoours Semiar) Lakehead

More information

Universal coding for classes of sources

Universal coding for classes of sources Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric

More information

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error STA 2023 Practice Questios Exam 2 Chapter 7- sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (-1) oe p ( 1 p) CI: prop.

More information

I. Why is there a time value to money (TVM)?

I. Why is there a time value to money (TVM)? Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Groups of diverse problem solvers can outperform groups of high-ability problem solvers

Groups of diverse problem solvers can outperform groups of high-ability problem solvers Groups of diverse problem solvers ca outperform groups of high-ability problem solvers Lu Hog ad Scott E. Page Michiga Busiess School ad Complex Systems, Uiversity of Michiga, A Arbor, MI 48109-1234; ad

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

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

More information

Confidence intervals and hypothesis tests

Confidence intervals and hypothesis tests Chapter 2 Cofidece itervals ad hypothesis tests This chapter focuses o how to draw coclusios about populatios from sample data. We ll start by lookig at biary data (e.g., pollig), ad lear how to estimate

More information

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test) No-Parametric ivariate Statistics: Wilcoxo-Ma-Whitey 2 Sample Test 1 Ma-Whitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo-) Ma-Whitey (WMW) test is the o-parametric equivalet of a pooled

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

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

CS103X: Discrete Structures Homework 4 Solutions

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

More information

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

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

More information