Convention Paper 6764

Size: px
Start display at page:

Download "Convention Paper 6764"

Transcription

1 Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or cosideratio by the Review Board. The AES takes o resposibility for the cotets. Additioal papers may be obtaied by sedig request ad remittace to Audio Egieerig Society, 60 East 4 d Street, New York, New York , USA; also see All rights reserved. Reproductio of this paper, or ay portio thereof, is ot permitted without direct permissio from the Joural of the Audio Egieerig Society. Optimisatio of Co-cetred Rigid ad Ope Spherical Microphoe Arrays Abhaya Parthy 1, Craig Ji, ad Adré va Schaik 1 School of Iformatio Techology, The Uiversity of Sydey, NSW, 006, Australia School of Electrical ad Iformatio Egieerig, The Uiversity of Sydey, NSW, 006, Australia {craig, ABSTRACT We preset a ovel microphoe array that cosists of a ope spherical array with a smaller rigid spherical array at its cetre. The distributio of microphoes, which results i the array havig the largest frequecy rage, for a give beamformig order, was obtaied by aalysig microphoe errors. For a fixed umber of microphoes, the results for several examples idicate that the maximum frequecy rage is obtaied whe the microphoes are relatively evely distributed betwee the ope ad rigid spheres. 1. INTRODUCTION May spherical microphoe array cofiguratios, such as that preseted by Meyer ad Elko [1] ad Abhayapala ad Ward [], have a limited useable frequecy rage, typically 3-4 octaves depedig o the umber of microphoes used. This frequecy rage limitatio is due to spatial aliasig at the high frequecies ad microphoe positioig errors at the low frequecies. For a fixed umber of microphoes o a sphere, spatial aliasig ca be reduced by reducig the spacig betwee microphoes ad by reducig the radius of the sphere. The trade off i reducig the radius of the sphere, however, is that the microphoe positioig error icreases due to the smaller size of the sphere. The spatial aliasig error ad the microphoe positioig error for a give arragemet of microphoes o a spherical microphoe array is depedat o the dimesioless parameter kr, where k is the wave umber ad r is the radius of the sphere. Utilisig multiple arrays of differig radii is a techique which allows a larger frequecy rage to be covered. Gover [3] uses two ope spherical microphoe arrays to capture a larger frequecy rage, a smaller spherical microphoe array for capturig high frequecies ad a larger spherical microphoe array for capturig low frequecies. Multiple ope spherical microphoe arrays ca be cetred at the same poit allowig the soud

2 Broadbad Spherical Microphoe Arrays field to be recorded at oe locatio, however, ope spherical microphoe arrays are disadvataged i that their error rises dramatically for certai values of kr. Rigid spherical microphoe arrays do ot have this problem [4]. It is preferable to use rigid spherical microphoe arrays whe recordig a soud field for this reaso. However, multiple rigid spherical microphoe arrays ca ot be cetred at the same poit, ad usig multiple rigid spherical microphoe arrays for recordig a soud field at locatios close to each other is ot practical, as the rigid spheres will scatter soud ad affect the soud field beig recorded by the other rigid spherical microphoe arrays. We preset a spherical microphoe array cofiguratio, which we have ot see reported previously i the literature, with microphoes mouted o both a ope ad rigid sphere with a commo cetre. The smaller rigid spherical microphoe array is used for recordig high frequecies, while the larger ope spherical microphoe array is used for recordig low frequecies. The soud scattered by the cetral rigid sphere ca be calculated aalytically at ay poit surroudig the sphere, ad thus the soud field at the surface of the ope sphere is kow [1]. This cofiguratio allows soud field recordig at oe locatio while retaiig the advatages of the rigid spherical microphoe array. Buildig a spherical microphoe array with this cofiguratio, usig a fixed umber of microphoes, requires that a umber of microphoes be placed o the rigid spherical microphoe array ad the remaiig microphoes be placed o the ope spherical microphoe array. I additio, the frequecy rages that the two spherical microphoe arrays cover must overlap. We preset a optimisatio algorithm that calculates the umber of microphoes that should be placed o the rigid ad ope spherical microphoe arrays to maximise the frequecy rage of the combied arrays for a give spherical harmoic order of the beamformer.. METHODS The optimisatio program was writte usig the MATLAB software eviromet. The useable kr rage is defied as the kr rage for the spherical microphoe array for which the microphoe positioig error ad the spatial aliasig error remai below a fixed value. The iputs for the optimisatio algorithm are the total umber of microphoes, the maximum tolerable sigal error, due to microphoe positioig error ad spatial aliasig error, i the spherical microphoe array, expressed as a oise-to-sigal ratio, the rage for the uiform distributio of the radom microphoe placemet error for the rigid ad ope spherical microphoe arrays, ad the spherical harmoic order of the beamformer. The optimisatio algorithm calculates the umber of microphoes that should be placed o the rigid ad ope spheres, the ratio of the ope sphere radius to the rigid sphere radius, ad the useable kr rage for the ope array ad the useable kr rage for the rigid array. Several assumptios were made i the desig of the optimisatio algorithm ad are detailed i the followig paragraphs. Firstly, we assume that the spherical harmoic order for the beamformer remais costat to esure reasoably costat gai across the spherical microphoe array s useable kr rage. The gai of the microphoe array is related to the directivity idex which is defied as the peak-to-average ratio of the beam patter expressed i decibels [3]. For spherical microphoe arrays processed usig spherical harmoics, the directivity idex is related to the spherical harmoic order of the beamformer ad icreases as the spherical harmoic order of the beamformer is icreased. The directivity idex remais relatively costat for a large rage of kr values. By oly beamformig at oe order o both arrays, the directivity idex will remai approximately costat across the etire useable frequecy rage. A secod assumptio is that the microphoes will be arraged o the ope ad rigid spheres with a early uiform spatial samplig scheme []. Nearly uiform spatial samplig schemes have bee show to be the most efficiet i terms of the umber of microphoes required [4]. I additio, oly spatial samplig schemes that satisfy the discrete spherical harmoic orthoormality criterio (see [4]) are used: m m α Y ( Ω ) Y ( Ω ) = δ δmm + ε mm, (1) AES 10th Covetio, Paris, Frace, 006 May 0 3 Page of 6

3 Broadbad Spherical Microphoe Arrays where Ω = ( θ, ϕ ) are the sample positios o a uit sphere i spherical coordiates, α are the weights for those sample positios, Y m is the spherical harmoic fuctio of order ad mode m, δ is the Kroecker delta fuctio, deotes complex cougatio, ad ε mm deotes the error i the sum for,, mm,. Oly spatial samplig positio lists which satisfy the spherical harmoic orthoormality criterio with 6 ε mm 3 10 for all,, mm, such that, N, where N is the spherical harmoic order of the beamformer, are used for arragig the microphoes o the spherical microphoe arrays. Several spatial samplig positio lists exist that do ot satisfy the orthoormality criterio (1) at a specified order, although there are other samplig positio lists, with fewer positios, that do satisfy the criterio at the same order. Spatial samplig positio lists that do ot satisfy the criterio are replaced with a spatial samplig positio list, with a lower umber of positios, which does satisfy the criterio. For example, at 4th order, we have foud spatial samplig positio lists with 37, 38, 39 ad 41 positios that do ot satisfy the orthoormality criterio, but a spatial samplig positio list with 36 positios that does satisfy the criterio. Fially, we also assume that measuremet oise is idepedet of the cofiguratio of the spherical microphoe array ad do ot iclude it i our sigal error calculatios..1. Optimisatio Algorithm The optimisatio algorithm begis with all microphoes cosidered to be o the rigid spherical microphoe array. For each iteratio of the algorithm, the umber of microphoes o the rigid spherical microphoe array decreases by oe ad the umber of microphoes o the ope spherical microphoe array icreases by oe. This iteratio cotiues util all microphoes are o the ope spherical microphoe array. For each iteratio, the sum of the microphoe positioig error ad the spatial aliasig error, herei referred to as PA error, for both the ope ad rigid spherical microphoe arrays is computed. The PA error is calculated assumig a sigle far-field, plae-wave source ad that the beamformer is steered i the directio of the icomig plae-wave. The spatial aliasig error, E a, is due to spatial samplig of the soud field o the surface of the spherical microphoe array. Spatial samplig limits the order to which a soud field ca be decomposed o the surface of the sphere ad the spherical harmoic decompositio of the soud field is trucated. The spatial aliasig error [4] is defied as E a = N = 0 = N+ 1 M = 1 b b 4π 4π α P (cos Θ ) P (cos Θ ) y s, () where N is the beamformig order, M is the umber of microphoes, Θ is the agle betwee the icomig plae wave ad the samplig positio Ω, s y is the 4 sigal power, ( N + 1) (4 π ), ad b is defied as where, ( ka) h' ( ka) 4 πi ( ( kr) h ( kr)), (3) h are the spherical Bessel ad Hakel fuctios respectively,, h are their derivatives, i = 1, ad a r is the radius of the cetral rigid sphere. The spatial aliasig error is depedat o the beamformig directio, thus it is calculated for 6 icomig plae-wave directios, distributed aroud the sphere as i [6], ad the averaged. It was foud, empirically that after averagig across 6 icomig wave directios the spatial aliasig error varied isigificatly as more icomig wave directios were added ad averaged. The microphoe positioig error, E Ω, is due to errors i the placemet of the microphoes o the spherical microphoe array. The microphoe placemet error, Δ, is the deviatio from the ideal microphoe positio, Ω, to the positio, Ω, give by θ = θ +Δ ad ϕ = ϕ +Δ siθ. (4) The microphoe placemet error, Δ, is assumed to be uiformly distributed such that Δ 0.00 radias. This rage of microphoe placemet error seemed reasoable give the size of the spherical microphoe array. Microphoe positioig error [4] is defied as AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 3 of 6

4 Broadbad Spherical Microphoe Arrays E Ω where = N = 0 = 0 M = 1 b b 4π 4π α P (cos Θ )[ P (cos Θ ) P (cos Θ )] y s,() Θ is the agle betwee the specified microphoe positio ad the icomig wave, ad Θ is the agle betwee the actual microphoe positio ad the icomig wave. The microphoe positioig error is depedat o the directio of beamformig, so a average error is calculated across the sphere for a umber of beamformig directios. The positioig error for the spherical microphoe array is calculated for 100 realisatios of the radom microphoe placemet error for each of 11 icomig plae-wave directios, distributed aroud the sphere as i [6], ad the averaged. It was foud empirically that after averagig 100 realisatios of the microphoe placemet error for each of 11 icomig wave directios the error varied isigificatly as more realisatios of the placemet error ad icomig wave directios were added ad averaged. For each iteratio, the PA error with the specified maximum tolerable oise-to-sigal ratio is used to calculate: firstly, the useable kr rage for the rigid spherical microphoe array, secodly, the largest ratio of the radius of the ope sphere to the radius of the rigid sphere, for which the PA error is less tha the maximum oise-to-sigal ratio ad such that the highest kr value for the ope spherical microphoe array is idetical to the lowest kr value for the rigid spherical microphoe array, fially, the useable kr rage for the ope spherical microphoe array. It should be oted that the useable kr rage for the ope spherical microphoe array is computed so that the largest value of kr lies before the first local maximum, i the alias error curve, which is greater tha the specified oise-to-sigal ratio. As show i Fig. 1, the spatial aliasig error curve for the ope sphere cotais umerous rages of kr for which the error rises dramatically. Noise-to-Sigal Ratio (db) kr Figure 1: Spatial aliasig error curve is show averaged over 6 icomig wave directios, for ope spherical array with 6 microphoes, havig a radius 7 times greater tha the rigid sphere located at its cetre. 3. RESULTS The optimisatio algorithm was executed to fid the optimal distributio for rigid-ope spherical microphoe array cofiguratios with 96, 64, 3 ad 4 microphoes. All spherical microphoe array cofiguratios were desiged to have a maximum error oise-to-sigal The first spherical microphoe array cofiguratio cosists of 96 microphoes operatig at 4th order. A miimum of 36 microphoes are required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. Thus, whe the rigid or ope spherical microphoe array cotais less tha 36 microphoes, that spherical microphoe array caot be used for beamformig. Referrig to Fig., the frequecy (i.e., kr) rage, for this cofiguratio, is at a maximum of.06 octaves whe 46 microphoes are placed o the ope spherical microphoe array ad 0 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is.66. The frequecy rage curve show i Fig. is ot smooth whe plotted agaist the umber of AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 4 of 6

5 Broadbad Spherical Microphoe Arrays Frequecy Rage (Octaves) Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure : The frequecy rage is show for a microphoe array cofiguratio with 96 microphoes ad workig to 4th order with maximum oise-to-sigal microphoes. This is caused by the spatial aliasig error which is highly o-liear across kr ad chages i a o-liear fashio as the umber of microphoes are icreased or reduced. The frequecy rage curves show i Figs. 3, 4 ad are ot smooth for the same reaso. The secod spherical microphoe array cofiguratio cosists of 64 microphoes operatig at 3rd order. A miimum of 6 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage for this cofiguratio is at a maximum of 6.3 octaves whe 3 microphoes are placed o the ope spherical microphoe array ad 3 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is 7.3. The third spherical microphoe array cofiguratio cosists of 3 microphoes operatig at d order. A miimum of 1 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage, for this cofiguratio, is at a maximum of octaves whe 16 microphoes are placed o the ope spherical microphoe array ad 16 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is Number of Microphoes o Rigid Sphere Figure 3: The frequecy rage is show for a microphoe array cofiguratio with 64 microphoes, ad workig to 3rd order with maximum oise-to-sigal Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure 4: The frequecy rage is show for a microphoe array cofiguratio with 3 microphoes ad workig to d order with maximum oise-tosigal The fourth spherical microphoe array cofiguratio cosists of 4 microphoes operatig at d order. A miimum of 1 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage, for this cofiguratio, is at a maximum of 9.90 octaves whe 1 microphoes are placed o the ope spherical microphoe array ad 1 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is 3.7. AES 10th Covetio, Paris, Frace, 006 May 0 3 Page of 6

6 Broadbad Spherical Microphoe Arrays Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure : The frequecy rage is show for a microphoe array cofiguratio with 4 microphoes ad workig to d order with maximum oise-tosigal 3.1. Example Microphoe Array A example spherical microphoe array is discussed to illustrate the practical cosideratios that are required whe costructig a spherical microphoe array. The example spherical microphoe array cofiguratio has 3 microphoes o the ope sphere ad 3 microphoes o the rigid sphere, ad has the highest frequecy rage of all the cofiguratios with 64 microphoes. The kr rage for the rigid microphoe array is , ad the kr rage for the ope microphoe array is First of all, the radius is related to f the frequecy by krc r = π f, (6) where c is the speed of soud. Thus, if we would like the array to work to a maximum frequecy of 14.0 khz, the radius of the rigid sphere would have to be 1.87 cm ad the radius of the ope sphere would the be 14.1cm. With this radius, the ope spherical microphoe array ca work to a low frequecy limit of 17 Hz. However, whe buildig the rigid microphoe array usig DPA type 4060-BM microphoes, which have a diameter of 0.4 mm ad a height 1.7 mm, it is ot be possible to place 3 microphoes o a sphere of radius 1.87 cm ad the radius of the rigid sphere has to be icreased to accommodate 3 microphoes. We fid that a sphere with a radius of at least.04 cm has to be used to accommodate 3 microphoes. With this radius for the rigid sphere, the high frequecy limit of the rigid spherical microphoe array becomes 1.8 khz. The radius of the ope sphere the becomes 1.4 cm, ad the low frequecy limit of the ope spherical microphoe array is 160 Hz. 4. CONCLUSION From the results preseted above, it is evidet that the largest useable frequecy rage for a cocetric rigid/ope spherical microphoe array beamformer that operates to a costat order is achieved whe microphoes are placed both o the rigid sphere ad the ope sphere. The results idicate that a relatively eve distributio of microphoes, betwee the ope ad rigid spheres, produces the highest frequecy rage.. REFERENCES [1] J. Meyer ad G. W. Elko, A highly scalable spherical microphoe array based o a orthoormal decompositio of the soudfield, i Proc. ICASSP, vol. II, 00, pp [] T. D. Abhayapala ad D. B. Ward, Theory ad desig of high order soud field microphoes usig spherical microphoe array, i Proc. ICASSP, vol. II, 00, pp [3] B. N. Gover, J. G. Rya, ad M. R. Stiso, Microphoe array measuremet system for aalysis of directioal ad spatial variatios of soud fields, J. Acoust. Soc. Amer., vol. 11, o., pp , 00. [4] B. Rafaely, Aalysis ad desig of spherical microphoe arrays, IEEE Tras. o Speech ad Audio Processig, vol. 13, o. 1, pp , 00. [] R. H. Hardi ad N. J. A. Sloae, McLare s improved sub cube ad other ew spherical desigs i three dimesios, Discrete Computatioal Geometry, vol. 1, pp , 199. [6] J. Fliege ad U. Maier, A two-stage approach for computig cubature formulae for the sphere, Ergebisberichte Agewadte Mathematik, No. 139T. Fachbereich Mathematik, Uiversität Dortmud, 441 Dortmud, Germay. September AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 6 of 6

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

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

NPTEL STRUCTURAL RELIABILITY

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

More information

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

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

Research Article Sign Data Derivative Recovery

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

More information

Basic Measurement Issues. Sampling Theory and Analog-to-Digital Conversion

Basic Measurement Issues. Sampling Theory and Analog-to-Digital Conversion Theory ad Aalog-to-Digital Coversio Itroductio/Defiitios Aalog-to-digital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results

More information

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

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

More information

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

Domain 1: Designing a SQL Server Instance and a Database Solution

Domain 1: Designing a SQL Server Instance and a Database Solution Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a

More information

An antenna is a transducer of electrical and electromagnetic energy. electromagnetic. electrical

An antenna is a transducer of electrical and electromagnetic energy. electromagnetic. electrical Basic Atea Cocepts A atea is a trasducer of electrical ad electromagetic eergy electromagetic electrical electrical Whe We Desig A Atea, We Care About Operatig frequecy ad badwidth Sometimes frequecies

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

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

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

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which

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

sum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by

sum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by Statistics Module Revisio Sheet The S exam is hour 30 miutes log ad is i two sectios Sectio A 3 marks 5 questios worth o more tha 8 marks each Sectio B 3 marks questios worth about 8 marks each You are

More information

1 Correlation and Regression Analysis

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

More information

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

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal Advaced Sciece ad Techology Letters, pp.31-35 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phase-locked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College

More information

Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics deals with the description or simple analysis of population or sample data. Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

More information

Volatility of rates of return on the example of wheat futures. Sławomir Juszczyk. Rafał Balina

Volatility of rates of return on the example of wheat futures. Sławomir Juszczyk. Rafał Balina Overcomig the Crisis: Ecoomic ad Fiacial Developmets i Asia ad Europe Edited by Štefa Bojec, Josef C. Brada, ad Masaaki Kuboiwa http://www.hippocampus.si/isbn/978-961-6832-32-8/cotets.pdf Volatility of

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

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

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

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

More information

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:

More information

AQA STATISTICS 1 REVISION NOTES

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

More information

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

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

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

Systems Design Project: Indoor Location of Wireless Devices

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

More information

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

Notes on Hypothesis Testing

Notes on Hypothesis Testing Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter

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

This is arithmetic average of the x values and is usually referred to simply as the mean.

This is arithmetic average of the x values and is usually referred to simply as the mean. prepared by Dr. Adre Lehre, Dept. of Geology, Humboldt State Uiversity http://www.humboldt.edu/~geodept/geology51/51_hadouts/statistical_aalysis.pdf STATISTICAL ANALYSIS OF HYDROLOGIC DATA This hadout

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

8.1 Arithmetic Sequences

8.1 Arithmetic Sequences MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

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

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

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

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

More information

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

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

More information

TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004

TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004 TIEE Teachig Issues ad Experimets i Ecology - Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013

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

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

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

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the

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

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

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC. SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol

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

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

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

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

Chair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics

Chair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to

More information

Overview on S-Box Design Principles

Overview on S-Box Design Principles Overview o S-Box Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA -721302 What is a S-Box? S-Boxes are Boolea

More information

CS100: Introduction to Computer Science

CS100: Introduction to Computer Science I-class Exercise: CS100: Itroductio to Computer Sciece What is a flip-flop? What are the properties of flip-flops? Draw a simple flip-flop circuit? Lecture 3: Data Storage -- Mass storage & represetig

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

3. Covariance and Correlation

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

More information

Domain 1 - Describe Cisco VoIP Implementations

Domain 1 - Describe Cisco VoIP Implementations Maual ONT (642-8) 1-800-418-6789 Domai 1 - Describe Cisco VoIP Implemetatios Advatages of VoIP Over Traditioal Switches Voice over IP etworks have may advatages over traditioal circuit switched voice etworks.

More information

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

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

More information

Lecture 2: Karger s Min Cut Algorithm

Lecture 2: Karger s Min Cut Algorithm priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.

More information

Capacity of Wireless Networks with Heterogeneous Traffic

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

More information

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

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

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

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

Cluster Validity Measurement Techniques

Cluster Validity Measurement Techniques Cluster Validity Measuremet Techiques Ferec Kovács, Csaba Legáy, Attila Babos Departmet of Automatio ad Applied Iformatics Budapest Uiversity of Techology ad Ecoomics Goldma György tér 3, H- Budapest,

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Example Consider the following set of data, showing the number of times a sample of 5 students check their  per day: Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day

More information

A Guide to the Pricing Conventions of SFE Interest Rate Products

A Guide to the Pricing Conventions of SFE Interest Rate Products A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios

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

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

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

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

Problem Set 1 Oligopoly, market shares and concentration indexes

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

More information

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

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

More information

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

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

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

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

Engineering Data Management

Engineering Data Management BaaERP 5.0c Maufacturig Egieerig Data Maagemet Module Procedure UP128A US Documetiformatio Documet Documet code : UP128A US Documet group : User Documetatio Documet title : Egieerig Data Maagemet Applicatio/Package

More information

Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.

Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median. Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable:

More information

Estimating Probability Distributions by Observing Betting Practices

Estimating Probability Distributions by Observing Betting Practices 5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,

More information

Quadrat Sampling in Population Ecology

Quadrat Sampling in Population Ecology Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may

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

On Formula to Compute Primes. and the n th Prime

On Formula to Compute Primes. and the n th Prime Applied Mathematical cieces, Vol., 0, o., 35-35 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih Abdul-Nabi

More information

Unit 20 Hypotheses Testing

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

More information

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

An Area Computation Based Method for RAIM Holes Assessment

An Area Computation Based Method for RAIM Holes Assessment Joural of Global Positioig Systems (2006) Vol. 5, No. 1-2:11-16 A Area Computatio Based Method for RAIM Holes Assessmet Shaoju Feg, Washigto Y. Ochieg ad Raier Mautz Cetre for Trasport Studies, Departmet

More information

3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average

3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average 5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives

More information

Forecasting techniques

Forecasting techniques 2 Forecastig techiques this chapter covers... I this chapter we will examie some useful forecastig techiques that ca be applied whe budgetig. We start by lookig at the way that samplig ca be used to collect

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

Designing Incentives for Online Question and Answer Forums

Designing Incentives for Online Question and Answer Forums Desigig Icetives for Olie Questio ad Aswer Forums Shaili Jai School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA shailij@eecs.harvard.edu Yilig Che School of Egieerig ad Applied

More information

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

More information

User manual and pre-programmed spreadsheets for performing revision analysis

User manual and pre-programmed spreadsheets for performing revision analysis User maual ad pre-programmed spreadsheets for performig revisio aalysis This documet describes how to perform revisio aalysis usig pre-programmed template spreadsheets based o data extracted from the OECD

More information

Chapter XIV: Fundamentals of Probability and Statistics *

Chapter XIV: Fundamentals of Probability and Statistics * Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive

More information

, a Wishart distribution with n -1 degrees of freedom and scale matrix.

, a Wishart distribution with n -1 degrees of freedom and scale matrix. UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that

More information