Distributions of Order Statistics

Size: px
Start display at page:

Download "Distributions of Order Statistics"

Transcription

1 Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1 dt B(a, b) 0 deotes the icomplete Beta fuctio. The probability desity fuctio of X k, is give as follows: f k: (x)=! (k 1)!( k)! (F(x))k 1 (1 F(x)) k f (x), where f is a populatio desity fuctio. The joit desity fuctio of order statistics X 1,, X 2,,...,X, has the form! f (x k ), < x 1 < x 2 < < x <, ad f 1,2,...,: (x 1,x 2,...,x )= 0, otherwise. There are some simple formulae for distributio of maxima ad miima. Example 2.1. Let radom variables X 1, X 2,...,X have a joit d.f. H(x 1,x 2,...,x )=P{X 1 x 1, X 2 x 2,..., X x }. The d.f. of M()=max{X 1, X 2,...,X } has the form P{M() x} = P{X 1 x, X 2 x,..., X x} = H(x,x,...,x). (2.1) Similarly we ca get the distributio of m()=mi{x 1, X 2,...,X }. Oe has P{m() x} = 1 P{m() > x} = 1 P{X 1 > x, X 2 > x,..., X > x}. (2.2) M. Ahsaullah et al., A Itroductio to Order Statistics, Atlatis Studies i Probability ad Statistics 3, DOI: / _2, Ó Atlatis Press

2 16 A Itroductio to Order Statistics Exercise 2.1. Fid the joit distributio fuctio of M( 1) ad M(). Exercise 2.2. Express d.f. of Y = mi{x 1,X 2 } i terms of joit d.f. H(x 1,x 2 )=P{X 1 x 1, X 2 x 2 }. Exercise 2.3. Let H(x 1,x 2 ) be the joit d.f. of X 1 ad X 2. Fid the joit d.f. of Y = mi{x 1,X 2 } ad Z = max{x 1,X 2 }. From (2.1) ad (2.2) oe obtais the followig elemetary expressios for the case, whe X 1, X 2,...,X preset a sample from a populatio d.f. F (ot ecessary cotiuous): ad P{M() x} = F (x) (2.3) P{m() x} = 1 (1 F(x)). (2.4) Exercise 2.4. Let X 1, X 2,...,X be a sample of size from a geometrically distributed radom variable X, such that P{X = m} =(1 p)p m, m = 0, 1, 2,... Fid P{Y r, Z < s}, r < s, where Y = mi{x 1, X 2,...,X } ad Z = max{x 1, X 2,...,X }. There is o difficulty to obtai d.f. s for sigle order statistics X k,.let F k, (x)=p{x k, x}. We see immediately from (2.3) ad (2.4) that F, (x)=f (x) ad F 1: (x)=1 (1 F(x)). The geeral formula for F k: (x) is ot much more complicated. I fact, F k: (x)=p{ at least k variables amog X 1, X 2,...,X are less or equal x} = m=k ( = m=k m Exercise 2.5. Prove that idetity P{ exactly m variables amog X 1, X 2,...,X are less or equal x} ) (F(x)) m (1 F(x)) m, 1 k. (2.5) m=k( m holds for ay 0 x 1, where ) x m (1 x) m = I x (k, k + 1) (2.6) I x (a,b)= 1 B(a, b) x 0 t a 1 (1 t) b 1 dt (2.7)

3 Distributios of order statistics 17 is the icomplete Beta fuctio with parameters a ad b, B(a,b) beig the classical Beta fuctio. By comparig (2.5) ad (2.6) oe obtais that F k: (x)=i F(x) (k, k + 1). (2.8) Remark 2.1. It follows from (2.8) that X k, has the beta distributio with parameters k ad k + 1, if X has the uiform o [0,1] distributio. Remark 2.2. Equality (2.8) is valid for ay distributio fuctio f. Remark 2.3. If we have tables of the fuctio I x (k, k + 1), it is possible to obtai d.f. F k: (x) for arbitrary d.f. F. Exercise 2.6. Fid the joit distributio of two order statistics X r, ad X s,. Example 2.2. Let us try to fid the joit distributio of all elemets of the variatioal series X 1,, X 2,,...,X,. It seems that the joit d.f. F 1,2,...,: (x 1,x 2,...,x )=P{X 1, x 1, X 2, x 2,..., X, x } promises to be very complicated. Hece, we cosider probabilities P(y 1,x 1,y 2,x 2,...,y,x )=P{y 1 < X 1, x 1, y 2 < X 2, x 2,..., y < X, x } for ay values y 1 < x 1 y 2 < x 2 y < x. It is evidet, that the evet A = {y 1 < X 1, x 1, y 2 < X 2, x 2,..., y < X, x } is a uio of! disjoit evets A(α(1),α(2),...,α()) = {y 1 < X α(1) x 1,y 2 < X α(2) x 2,...,y < X α() x },where the vector (α(1),α(2),...,α()) rus all permutatios of umbers 1, 2,...,. The symmetry argumet shows that all evets A(α(1), α(2),..., α()) have the same probability. Note that this probability is equal to Fially, we obtai that (F(x k ) F(y k )). P{y 1 < X 1, x 1, y 2 < X 2, x 2,..., y < X, x } =! (F(x k ) F(y k )). (2.9)

4 18 A Itroductio to Order Statistics Let ow cosider the case whe our populatio distributio has a desity fuctio f.it meas that for almost all X (i.e., except, possibly, a set of zero Lebesgue measure) F (x)= f (x). I this situatio (2.9) eables us to fid a expressio for the joit probability desity fuctio (p.d.f.) (deote it f 1,2,..., (x 1,x 2,...,x )) of order statistics X 1,, X 2,,...,X,.I fact, differetiatig both sides of (2.9) with respect to x 1, x 2,...,x, we get the importat equality f 1,2,...,: (x 1,x 2,...,x )=! f (x k ), < x 1 < x 2 < < x <. (2.10) Otherwise (if iequalities x 1 < x 2 < < x fail) we aturally put f 1,2,...,: (x 1,x 2,...,x )=0. Takig (2.10) as a startig poit oe ca obtai differet results for joit distributios of arbitrary sets of order statistics. Example 2.3. Let f m: deote the p.d.f. of X m,. We get from (2.10) that f m: (x)= f 1,2,...,: (x 1,...,x k 1,x,x k+1,...,x )dx 1 dx k 1 dx k+1 dx =! f (x) m 1 f (x k ) where the itegratio is over the domai k=m+1 f (x k )dx 1 dx m 1 dx m+1 dx, (2.11) < x 1 < < x m 1 < x < x m+1 < < x <. The symmetry of m 1 f (x k ) with respect to x 1,...,x m 1,aswellasthesymmetryof f (x k ) k=m+1 with respect to x m+1,...,x, helps us to evaluate the itegral o the RHS of (2.11) as follows: m 1 = f (x k ) 1 (m 1)! k=m+1 m 1 x = (F(x))m 1 (1 F(x)) m (m 1)!( m)! f (x k )dx 1 dx m 1 dx m+1 dx 1 f (x k )dx k ( m)! k=m+1 x f (x k )dx k. (2.12)

5 Distributios of order statistics 19 Combiig (2.11) ad (2.12), we get that! f m: (x)= (m 1)!( m)! (F(x))m 1 (1 F(x)) m f (x). (2.13) Ideed, equality (2.13) is immediately follows from the correspodig formula for d.f. s of sigle order statistics (see (2.8), for example), but the techique, which we used to prove (2.13), is applicable for more complicated situatios. The followig exercise ca illustrate this statemet. Exercise 2.7. Fid the joit p.d.f. f k(1),k(2),...,k(r): (x 1,x 2,...,x r ) of order statistics X k(1),, X k(2),,...,x k(r),,where1 k(1) < k(2) < < k(r). Remark 2.4. I the sequel we will ofte use the particular case of the joit probability desity fuctios from exercise 2.7, which correspods to the case r = 2. It turs out that! f i, j: (x 1,x 2 )= (i 1)!( j i 1)!( j)! (F(x 1 )) i 1 (F(x 2 ) F(x 1 )) j i 1 (1 F(x 2 )) j f (x 1 ) f (x 2 ), (2.14) if 1 i < j ad x 1 < x 2. Expressio (2.10) eables us also to get the joit d.f. F 1,2,...,: (x 1,x 2,...,x )=P{X 1, x 1, X 2, x 2,..., X, x }. Oe has where F 1,2,...,: (x 1,x 2,...,x )=! D f (u k )du 1 du, (2.15) D = {U 1,...,U : U 1 < U 2 < < U ; U 1 < x 1, U 2 < x 2,..., U < x }. Note that (2.15) is equivalet to the expressio F 1,2,...,: (x 1,x 2,...,x )=! du 1 du, D (2.16) where itegratio is over D = {U 1,...,U : U 1 < U 2 < < U ; U 1 < F(x 1 ), U 2 < F(x 2 ),..., U < F(x )}. Remark 2.5. It ca be proved that ulike (2.15), which eeds the existece of populatio desity fuctio f, expressio (2.16), as well as (2.9), is valid for arbitrary distributio fuctio F.

6 20 A Itroductio to Order Statistics Check your solutios Exercise 2.1 (solutio). If x y,the Otherwise, P{M( 1) x, M() y} = P{M() y} = H(y,y,...,y). P{M( 1) x, M() y} = P{M( 1) x, X y} = H(x,...,x,y}. Exercise 2.2 (aswer). where ad Exercise 2.3 (solutio). If x y,the If x < y,the P{Y x} = 1 H(x, ) H(,x)+H(x,x), H(x, )=P{X 1 x, X 2 < } = P{X 1 x} H(,x)=P{X 2 x}. P{Y x, Z y} = P{Z y} = H(y,y). P{Y x, Z y} = P{X 1 x, X 2 y} + P{X 1 y, X 2 x} P{X 1 x, X 2 x} Exercise 2.4 (solutio). We see that = H(x,y)+H(y,x) H(x,x). P{Y r, Z < s} = P{r X k < s, k = 1, 2,...,} =(P{r X < s}) =(P{X r} P{X s}) =(p r p s ). Exercise 2.5 (solutio). It is easy to see that (2.6) is valid for x = 0. Now it suffices to prove that both sides of (2.6) have equal derivatives. The derivative of the RHS is aturally equal to x k 1 (1 x) k B(k, k + 1) =!xk 1 (1 x) k (k 1)!( k)!, because B(a, b) = Γ(a)Γ(b)/Γ(a + b) ad the gamma fuctio satisfies equality Γ(k) = (k 1)!fork = 1, 2,... It turs out (after some simple calculatios) that the derivative of the LHS also equals!x k 1 (1 x) k (k 1)!( k)!.

7 Distributios of order statistics 21 Exercise 2.6 (solutio). Let r < s. Deote F r,s: (x 1,x 2 )=P{X r, x 1, X s, x 2 }. If x 2 x 1, the evidetly P{X r, x 1, X s, x 2 } = P{X s, x 2 } ad ) F r,s: (x 1,x 2 )= (F(x 2 )) m=s( m (1 F(x 2 )) m = 1 m F(x2 )(s, s + 1). Cosider ow the case x 2 > x 1. To fid F r,s: (x 1,x 2 ) let us metio that ay X from the sample X 1, X 2,...,X (idepedetly o other X s) with probabilities F(x 1 ), F(x 2 ) F(x 1 ) ad 1 F(x 2 ) ca fall ito itervals (,x 1 ], (x 1,x 2 ], (x 2, ) respectively. Oe sees that the evet A = {X r, x 1, X s, x 2 } is a uio of some disjoit evets a i, j, i j = {i elemets of the sample fall ito (,x 1 ], j elemets fall ito iterval (x 1,x 2 ],ad ( i j) elemets lie to the right of x 2 }. Recallig the polyomial distributio we obtai that! P{A i, j, i j } = i! j!( i j)! (F(x 1)) i (F(x 2 ) F(x 1 )) j (1 F(x 2 )) i j. To costruct A oe has to take all a i, j, i j such that r i, j 0ads i + j. Hece, F r,s: (x 1,x 2 )=P{A} = = i i=r j=max{0,s i} i i=r j=max{0,s i} P{A i, j, i j }! i! j!( i j)! (F(x 1)) i (F(x 2 ) F(x 1 )) j (1 F(x 2 )) i j. Exercise 2.7 (aswer). Deote for coveiece, k(0)=0, k(r + 1)= + 1, x 0 = ad x r+1 =. The f k(1),k(2),...,k(r): (x 1,x 2,...,x r )=! r+1 (F(x r+1 m ) F(x m 1 )) k(m) k(m 1) 1 r f (x m ), m=1 m=1 (k(m) k(m 1) 1)! m=1 if x 1 < x 2 < < x r,ad 0, otherwise. I particular, if r = 2, 1 i < j,adx 1 < x 2,the! f i, j: (x 1,x 2 )= (i 1)!( j i 1)!( j)! (F(x 1 )) i 1 (F(x 2 ) F(x 1 )) j i 1 (1 F(x 2 )) j f (x 1 ) f (x 2 ).

8

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

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

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

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

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

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

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

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

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

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

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

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

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

More information

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

Elementary Theory of Russian Roulette

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

More information

Confidence Intervals 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

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

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

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

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

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

More information

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

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

More information

Section 11.3: The Integral Test

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

More information

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

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

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

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1) BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet

More information

3 Basic Definitions of Probability Theory

3 Basic Definitions of Probability Theory 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

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

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

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

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

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

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

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

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

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece;

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

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

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

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

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

3. Greatest Common Divisor - Least Common Multiple

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

More information

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

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

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

5.3. Generalized Permutations and Combinations

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

More information

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

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

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

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

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the

More information

Exploratory Data Analysis

Exploratory Data Analysis 1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

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

Lecture 4: Cheeger s Inequality

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

More information

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

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

More information

1 Computing the Standard Deviation of Sample Means

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

More information

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

Solving Logarithms and Exponential Equations

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

More information

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

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

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

Partial Di erential Equations

Partial Di erential Equations Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio

More information

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE 11 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November

More information

Permutations, the Parity Theorem, and Determinants

Permutations, the Parity Theorem, and Determinants 1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits

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

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

Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>

Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu> (March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

More information

Math 113 HW #11 Solutions

Math 113 HW #11 Solutions Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate

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

Plug-in martingales for testing exchangeability on-line

Plug-in martingales for testing exchangeability on-line Plug-i martigales for testig exchageability o-lie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk

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

Entropy of bi-capacities

Entropy of bi-capacities Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics

More information

Notes on exponential generating functions and structures.

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

More information

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

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

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

2-3 The Remainder and Factor Theorems

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

More information

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE 11 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a -page formula sheet. Please tur over Mathematics/P DoE/Exemplar

More information

Class Meeting # 16: The Fourier Transform on R n

Class Meeting # 16: The Fourier Transform on R n MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

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

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

How To Solve The Homewor Problem Beautifully

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

More information

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

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

More information

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

Multiplexers and Demultiplexers

Multiplexers and Demultiplexers I this lesso, you will lear about: Multiplexers ad Demultiplexers 1. Multiplexers 2. Combiatioal circuit implemetatio with multiplexers 3. Demultiplexers 4. Some examples Multiplexer A Multiplexer (see

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

On the L p -conjecture for locally compact groups

On the L p -conjecture for locally compact groups Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/030237-6, ublished olie 2007-08-0 DOI 0.007/s0003-007-993-x Archiv der Mathematik O the L -cojecture for locally comact

More information

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe

More information

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff, NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical

More information

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

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

More information

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

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

More information

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

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

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

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

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

Analysis Notes (only a draft, and the first one!)

Analysis Notes (only a draft, and the first one!) Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

More information

Metric, Normed, and Topological Spaces

Metric, Normed, and Topological Spaces Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolute-value metric

More information

Topic 5: Confidence Intervals (Chapter 9)

Topic 5: Confidence Intervals (Chapter 9) Topic 5: Cofidece Iterval (Chapter 9) 1. Itroductio The two geeral area of tatitical iferece are: 1) etimatio of parameter(), ch. 9 ) hypothei tetig of parameter(), ch. 10 Let X be ome radom variable with

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