A note on exact moments of order statistics from exponentiated log-logistic distribution
|
|
- Randolph Baker
- 8 years ago
- Views:
Transcription
1 ProbStat Forum, Volume 7, July 214, Pages ISSN ProbStat Forum is an e-ournal. For details please visit A note on exact moments of order statistics from exponentiated log-logistic distribution Haseeb Athar, Nayabuddin Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh-22 2, India Abstract. In this paper we establish explicit expressions for single and product moments of order statistics from exponentiated log-logistic distribution. These expressions are used to calculate the mean and variances. 1. Introduction Let X 1, X 2,..., X n be a random sample of size n from a continuous population having probability density function (pdf) f(x) and distribution function (df) F (x). Let X 1:n X 2:n... X n:n be the corresponding order statistics. The pdf of X r:n, the rth order statistic is given by David and Nagaraa (23) f r:n (x) = (r 1)!(n r)! [F (x)]r 1 [1 F (x)] n r f(x), < x <, (1) and oint pdf of X r:n, X s:n, 1 r < s n is given as f r,s:n (x, y) = (r 1)!(s r 1)!(n s)! [F (x)]r 1 [F (y) F (x)] s r 1 [1 F (y)] n s f(x)f(y), (2) < x < y <. A random variable X is said to have exponentiated log-logistic distribution (Rosaiah et al., 26) if its pdf is given by x f(x) = θx (+1)[ ] θ+1, θ, >, < x <, (3) 1 + x and the corresponding df of X is [ x ] θ, F (x) = θ, >, < x <. (4) 1 + x For more details on this distribution and its application one may refer to Rosaiah et al. (26, 27). Keywords. Order statistics, Exponentiated log-logistic distribution, Single and product moments. Received: 5 December 213; Accepted: 6 June 214 Corresponding author address: haseebathar@hotmail.com (Haseeb Athar )
2 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Moments of order statistics for some specific distributions are investigated by several authors in the literature. Malik (1966) derived the expression for exact moments of order statistics from Pareto distribution. Malik (1967) obtained the explicit expression for moments of order statistics of power function distribution. Khan and Khan (1987) obtained the moments of order statistics from Burr distribution. Khan and Ali (1995) derived the ratio and inverse moments of order statistic from Burr distribution. Further, Ali and Khan (1996) established the ratio and inverse moments of order statistic from Weibull and exponential distribution. In this paper we have obtained simple expressions for the exact moments of order statistic from exponentiated log-logistic distribution. Also means and variances are tabulated. 2. Single moments Lemma 2.1. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a and b, we have that ( I (a, ) = θb ) + θ(a + 1), 1, I (a, b) = Proof. From (5), we have I (a, ) = x [F (x)] a [1 F (x)] b f(x)dx. (5) x [F (x)] a f(x)dx = θ x 1[ ( ) θ x Set 1+x = t θ in (6), after simplification we get required result. x 1 + x ] θ(a+1)+1 dx. (6) Lemma 2.2. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a and b, we have that I (a, b) = b ( ) b ( 1) k I (a + k, ) = θ k k= b ( ) b ( ( 1) k B k ) + θ(a + k + 1), 1. Proof. It can be proved by expanding [1 F (x)] b binomially in (5) and using Lemma 2.1. k= Theorem 2.3. For exponentiated log-logistic distribution as given in (3) and 1 r n,, θ >, we have that n r ( n r n r ( ) n r ( E(Xr:n) = C r:n ( 1) )I k (r +k 1, ) = θc r:n ( 1) k B k k ) +θ(r +k), 1, (7) C r:n = k= (r 1)!(n r)!. Proof. From (1), we have E(X r:n) = (r 1)!(n r)! On application of Lemma 2.2 in (8), we get required result. k= x [F (x)] r 1 [1 F (x)] n r f(x)dx. (8)
3 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Identity 2.1. For 1 r n 1, we have that C r:n n r ( ) n r 1 ( 1) k k r + k k= = 1. (9) Proof. At = in (7), we have n r ( ) n r 1 1 = C r:n ( 1) k k r + k k= and hence the result given in (9). 3. Product moments Lemma 3.1. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a and c, we have that θ 2 ( i + ) I i, (a,, c) = I i, (a, b, c) = + θ(c + 1)B 3 F 2 x + θ(c + 1), + θ(c + 1) + 1, and a 1, a 2,..., a p [ p pf q, 1 = b 1, b 2,..., b q r= =1 + θ(a + c + 2), 1 i, + θ(a + c + 2) + 1 i+ + θ(a + c + 2) ; 1 x i y [F (x)] a [F (y) F (x)] b [F (y)] c f(x)f(y)dydx (1) Γ(a + r) ][ q Γ(b ) ]. a b =1 + r For p = q + 1 and q =1 b p =1 a >, see Mathai and Saxena (1973). Proof. We have from (1) I i, (a,, c) = x In view of (3), (4) and (11), we have By setting I i, (a,, c) = (θ) 2 ( y x i y [F (x)] a [F (y)] c f(x)f(y)dydx. (11) x i 1[ x ] θ(a+1)+1[ x 1 + x y 1[ y ] θ(c+1)+1 ] dy 1 + y 1+y ) θ = t θ in (12), we get after simplification I i, (a,, c) = θ 2 B x (p, q) = x x i 1[ t p 1 (1 t) q 1 dt, dx. (12) x ] θ(a+1)+1 ( B 1 + x x 1+x + θ(c + 1), 1 ) dx (13)
4 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages is the incomplete beta function. Also note that Mathai and Saxena (1973) proved that and B x (p, q) = xp p 2 F 1 (p, 1 q, p + 1; x) 1 t a 1 (1 t) b 1 2 F 1 (c, d ; e; t)dt = B(a, b) 3 F 2 (c, d, a ; e, a + b ; 1). Substituting these value in (13), we get the required result. Lemma 3.2. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a, b and c, we have that I i, (a, b, c) = = b ( ) b ( 1) k I i, (a + k,, b + c k) k k= θ 2 ( i + + θ(c + 1)B + θ(a + c + 2), 1 i ) + θ(c + 1),, 3 F 2 + θ(c + 1) + 1, + θ(a + c + 2) + 1 i+ + θ(a + c + 2) ; 1. Proof. Lemma can be proved by expanding [F (y) F (x)] b binomially in (1) and using Lemma 2.1. Theorem 3.3. For exponentiated log-logistic distribution as given in (2) and, θ > and 1 r < s n 1, we have that E(X i, n s s r 1 ( )( s r 1 n s r,s:n) = C r,s:n ( 1) k+l k l l= k= n s s r 1 ( )( ) s r 1 n s = C r,s:n ( 1) k+l k l l= k= ( i + B + θ(s + l + k), 1 i ) + θ(s + l r),, Proof. From (2), we have 3 F 2 E(X i, r,s:n) = C r,s:n x + θ(s + l r) + 1, On application of Lemma 3.2, (14) can be proved. o ) I i, (r + k 1,, s + l r 1) θ 2 + θ(s + l r) + θ(s + l + k) + 1 i+ + θ(s + l + k), 1 [F (x)] r 1 [F (y) F (x)] s r 1 [1 F (y)] n s f(x)f(y)dydx.. (14)
5 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Table 1: Mean of order statistics n r θ = 2, = 2 θ = 3, = 3 θ = 4, = 4 θ = 5, = Here in Table 1, it may be noted that the well known property of order statistics n i=1 E(X i:n) = ne(x) (David and Nagaraa, 23) is satisfied. Table 2: Variance of order statistics n r θ = 2, = 3 θ = 2, = 4 θ = 3, = 4 θ = 3, = References Ali, M.A., Khan, A. H. (1996) Ratio and inverse moments of order statistics from Weibull and exponential distribution, J. Applied statistical science 4(1), 1 7. David, H.A., Nagaraa H.N. (23) Order Statistics, John Wiley, New York. Khan, A.H., Khan, I.A. (1987) Moments of order statistics from Burr distribution and its characterizations, Metron XLV,
6 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Khan, A.H., Ali, M.A. (1995) Ratio and inverse moments of order statistics from Burr distribution, J. Ind. Soc. Prob. Statist. 2, Malik, H.J. (1966) Exact moments of order statistics from Pareto distribution, Skand. Aktuarie Tidskr 49, Malik, H.J. (1967) Exact moments of order statistics from a power function distribution, Skand. Aktuar. 5, Mathai, A.M., Saxena, R.K. (1973) Generalized hyper-geometric functions with applications in statistics and physical science, Lecture Notes in Mathematics 348, Springer-Verlag, Berlin. Rosaiah, K., Kantam, R.R.L., Santosh Kumar (26) Reliability test plan for exponentiated log-logistic distribution, Economic Quality Control 21, Rosaiah, K., Kantam, R.R.L., Santosh Kumar (27) Exponentiated log-logistic distribution an economic reliability test plan, Pakistan J. Statist. 23,
A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS
A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS Eusebio GÓMEZ, Miguel A. GÓMEZ-VILLEGAS and J. Miguel MARÍN Abstract In this paper it is taken up a revision and characterization of the class of
More informationUNIT I: RANDOM VARIABLES PART- A -TWO MARKS
UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationNonparametric adaptive age replacement with a one-cycle criterion
Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
More informationPermanents, Order Statistics, Outliers, and Robustness
Permanents, Order Statistics, Outliers, and Robustness N. BALAKRISHNAN Department of Mathematics and Statistics McMaster University Hamilton, Ontario, Canada L8S 4K bala@mcmaster.ca Received: November
More informationProbability Calculator
Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that
More informationINSURANCE RISK THEORY (Problems)
INSURANCE RISK THEORY (Problems) 1 Counting random variables 1. (Lack of memory property) Let X be a geometric distributed random variable with parameter p (, 1), (X Ge (p)). Show that for all n, m =,
More informationLecture Notes 1. Brief Review of Basic Probability
Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters -3 are a review. I will assume you have read and understood Chapters -3. Here is a very
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions
Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More informationMonitoring Software Reliability using Statistical Process Control: An MMLE Approach
Monitoring Software Reliability using Statistical Process Control: An MMLE Approach Dr. R Satya Prasad 1, Bandla Sreenivasa Rao 2 and Dr. R.R. L Kantham 3 1 Department of Computer Science &Engineering,
More informationA Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails
12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More informationNote on some explicit formulae for twin prime counting function
Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:
More informationPr(X = x) = f(x) = λe λx
Old Business - variance/std. dev. of binomial distribution - mid-term (day, policies) - class strategies (problems, etc.) - exponential distributions New Business - Central Limit Theorem, standard error
More informationSTAT 3502. x 0 < x < 1
Solution - Assignment # STAT 350 Total mark=100 1. A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous
More informationWHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION?
WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION? DAVID M MASON 1 Statistics Program, University of Delaware Newark, DE 19717 email: davidm@udeledu JOEL ZINN 2 Department of Mathematics,
More informationThe degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More informationSection 5.1 Continuous Random Variables: Introduction
Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,
More informationLecture 6: Discrete & Continuous Probability and Random Variables
Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September
More informationTests for exponentiality against the M and LM-classes of life distributions
Tests for exponentiality against the M and LM-classes of life distributions B. Klar Universität Karlsruhe Abstract This paper studies tests for exponentiality against the nonparametric classes M and LM
More informationLecture 13: Martingales
Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of
More information2WB05 Simulation Lecture 8: Generating random variables
2WB05 Simulation Lecture 8: Generating random variables Marko Boon http://www.win.tue.nl/courses/2wb05 January 7, 2013 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions Generating
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationMath 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions
Math 70, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the
More informationHow To Calculate The Power Of A Cluster In Erlang (Orchestra)
Network Traffic Distribution Derek McAvoy Wireless Technology Strategy Architect March 5, 21 Data Growth is Exponential 2.5 x 18 98% 2 95% Traffic 1.5 1 9% 75% 5%.5 Data Traffic Feb 29 25% 1% 5% 2% 5 1
More informationExponential Distribution
Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1
More informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
More informationFuzzy Probability Distributions in Bayesian Analysis
Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:
More informationThe Exponential Distribution
21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding
More informationPrinciple of Data Reduction
Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then
More informationTRANSACTIONS JUNE, 1969 AN UPPER BOUND ON THE STOP-LOSS NET PREMIUM--ACTUARIAL NOTE NEWTON L. BOWERS, JR. ABSTRACT
TRANSACTIONS OF SOCIETY OF ACTUARIES 1969 VOL. 21 PT. 1 NO. 60 VOL. XXI, PART I M~TINO No. 60 TRANSACTIONS JUNE, 1969 AN UPPER BOUND ON THE STOP-LOSS NET PREMIUM--ACTUARIAL NOTE NEWTON L. BOWERS, JR. ABSTRACT
More informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationCorrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk
Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk Jose Blanchet and Peter Glynn December, 2003. Let (X n : n 1) be a sequence of independent and identically distributed random
More information5. Continuous Random Variables
5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationA Model of Optimum Tariff in Vehicle Fleet Insurance
A Model of Optimum Tariff in Vehicle Fleet Insurance. Bouhetala and F.Belhia and R.Salmi Statistics and Probability Department Bp, 3, El-Alia, USTHB, Bab-Ezzouar, Alger Algeria. Summary: An approach about
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More informationProperties of Future Lifetime Distributions and Estimation
Properties of Future Lifetime Distributions and Estimation Harmanpreet Singh Kapoor and Kanchan Jain Abstract Distributional properties of continuous future lifetime of an individual aged x have been studied.
More informationManual for SOA Exam MLC.
Chapter 4. Life Insurance. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam MLC. Fall 29 Edition. available at http://www.actexmadriver.com/ c 29. Miguel A. Arcones.
More informationRacetrack Betting and Consensus of Subjective Probabilities
Racetrack Betting and Consensus of Subective Probabilities Lawrence D. Brown 1 and Yi Lin 2 University of Pennsylvania and University of Wisconsin, Madison Abstract In this paper we consider the dynamic
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche
More informationGenerating Random Samples from the Generalized Pareto Mixture Model
Generating Random Samples from the Generalized Pareto Mixture Model MUSTAFA ÇAVUŞ AHMET SEZER BERNA YAZICI Department of Statistics Anadolu University Eskişehir 26470 TURKEY mustafacavus@anadolu.edu.tr
More informationOn the number-theoretic functions ν(n) and Ω(n)
ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,
More informationWhat is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference
0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More informationOrder Statistics. Lecture 15: Order Statistics. Notation Detour. Order Statistics, cont.
Lecture 5: Statistics 4 Let X, X 2, X 3, X 4, X 5 be iid random variables with a distribution F with a range of a, b). We can relabel these X s such that their labels correspond to arranging them in increasing
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More informationCourse Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics
Course Notes for Math 16: Mathematical Statistics Approximation Methods in Statistics Adam Merberg and Steven J. Miller August 18, 6 Abstract We introduce some of the approximation methods commonly used
More informationMATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...
MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................
More informationVERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS
VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled
More informationRandom Variate Generation (Part 3)
Random Variate Generation (Part 3) Dr.Çağatay ÜNDEĞER Öğretim Görevlisi Bilkent Üniversitesi Bilgisayar Mühendisliği Bölümü &... e-mail : cagatay@undeger.com cagatay@cs.bilkent.edu.tr Bilgisayar Mühendisliği
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More information14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style
Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of
More informationECE302 Spring 2006 HW5 Solutions February 21, 2006 1
ECE3 Spring 6 HW5 Solutions February 1, 6 1 Solutions to HW5 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationStatistics 100A Homework 8 Solutions
Part : Chapter 7 Statistics A Homework 8 Solutions Ryan Rosario. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, the one-half
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationarxiv:math/0202219v1 [math.co] 21 Feb 2002
RESTRICTED PERMUTATIONS BY PATTERNS OF TYPE (2, 1) arxiv:math/0202219v1 [math.co] 21 Feb 2002 TOUFIK MANSOUR LaBRI (UMR 5800), Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationTransformations and Expectations of random variables
Transformations and Epectations of random variables X F X (): a random variable X distributed with CDF F X. Any function Y = g(x) is also a random variable. If both X, and Y are continuous random variables,
More informationMaximum Likelihood Estimation
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
More informationPROPERTECHNIQUEOFSOFTWARE INSPECTIONUSING GUARDED COMMANDLANGUAGE
International Journal of Computer ScienceandCommunication Vol. 2, No. 1, January-June2011, pp. 153-157 PROPERTECHNIQUEOFSOFTWARE INSPECTIONUSING GUARDED COMMANDLANGUAGE Neeraj Kumar Singhania University,
More informationThe Analysis of Data. Volume 1. Probability. Guy Lebanon
The Analysis of Data Volume 1 Probability Guy Lebanon First Edition 2012 Probability. The Analysis of Data, Volume 1. First Edition, First Printing, 2013 http://theanalysisofdata.com Copyright 2013 by
More informationApproximation of Aggregate Losses Using Simulation
Journal of Mathematics and Statistics 6 (3): 233-239, 2010 ISSN 1549-3644 2010 Science Publications Approimation of Aggregate Losses Using Simulation Mohamed Amraja Mohamed, Ahmad Mahir Razali and Noriszura
More informationSumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationErrata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page
Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8
More informationON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION
Statistica Sinica 17(27), 289-3 ON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION J. E. Chacón, J. Montanero, A. G. Nogales and P. Pérez Universidad de Extremadura
More informationA Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University
A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a
More informationPROBABILITY AND STATISTICS. Ma 527. 1. To teach a knowledge of combinatorial reasoning.
PROBABILITY AND STATISTICS Ma 527 Course Description Prefaced by a study of the foundations of probability and statistics, this course is an extension of the elements of probability and statistics introduced
More informationCOUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS
COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics
More informationInference on the parameters of the Weibull distribution using records
Statistics & Operations Research Transactions SORT 39 (1) January-June 2015, 3-18 ISSN: 1696-2281 eissn: 2013-8830 www.idescat.cat/sort/ Statistics & Operations Research c Institut d Estadstica de Catalunya
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Certain derivations presented in this course have been somewhat heavy on algebra. For example, determining the expectation of the Binomial distribution (page 5.1 turned
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationBipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES. 1. Introduction
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Bipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES Abstract. In this article the notion of acceleration convergence of double sequences
More informationON ROUGH (m, n) BI-Γ-HYPERIDEALS IN Γ-SEMIHYPERGROUPS
U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, 2013 ISSN 1223-7027 ON ROUGH m, n) BI-Γ-HYPERIDEALS IN Γ-SEMIHYPERGROUPS Naveed Yaqoob 1, Muhammad Aslam 1, Bijan Davvaz 2, Arsham Borumand Saeid 3 In this
More informationM2S1 Lecture Notes. G. A. Young http://www2.imperial.ac.uk/ ayoung
M2S1 Lecture Notes G. A. Young http://www2.imperial.ac.uk/ ayoung September 2011 ii Contents 1 DEFINITIONS, TERMINOLOGY, NOTATION 1 1.1 EVENTS AND THE SAMPLE SPACE......................... 1 1.1.1 OPERATIONS
More informationSingle-Period Balancing of Pay Per-Click and Pay-Per-View Online Display Advertisements
Single-Period Balancing of Pay Per-Click and Pay-Per-View Online Display Advertisements Changhyun Kwon Department of Industrial and Systems Engineering University at Buffalo, the State University of New
More informationA Tutorial on Probability Theory
Paola Sebastiani Department of Mathematics and Statistics University of Massachusetts at Amherst Corresponding Author: Paola Sebastiani. Department of Mathematics and Statistics, University of Massachusetts,
More informationLectures 5-6: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationSecond Order Differential Equations with Hypergeometric Solutions of Degree Three
Second Order Differential Equations with Hypergeometric Solutions of Degree Three Vijay Jung Kunwar, Mark van Hoeij Department of Mathematics, Florida State University Tallahassee, FL 0-07, USA vkunwar@mathfsuedu,
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More informationAlgorithms and Methods for Distributed Storage Networks 9 Analysis of DHT Christian Schindelhauer
Algorithms and Methods for 9 Analysis of DHT Institut für Informatik Wintersemester 2007/08 Distributed Hash-Table (DHT) Hash table does not work efficiently for inserting and deleting Distributed Hash-Table
More information( ) = 1 x. ! 2x = 2. The region where that joint density is positive is indicated with dotted lines in the graph below. y = x
Errata for the ASM Study Manual for Exam P, Eleventh Edition By Dr. Krzysztof M. Ostaszewski, FSA, CERA, FSAS, CFA, MAAA Web site: http://www.krzysio.net E-mail: krzysio@krzysio.net Posted September 21,
More information