Lecture 5: Probability Distributions. Random Variables
|
|
- Asher Wilkinson
- 5 years ago
- Views:
Transcription
1 Lecture 5: Probablty Dstrbutons Random Varables Probablty Dstrbutons Dscrete Random Varables Contnuous Random Varables and ther Dstrbutons Dscrete Jont Dstrbutons Contnuous Jont Dstrbutons Independent d Random Varables Summary Measures Moments of Condtonal and Jont Dstrbutons Correlaton and Covarance Random Varables A sample space s a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the real lne. It s gven by x : S R. Each element n the sample space has an assocated probablty and such probabltes sum or ntegrate to one. 1
2 Probablty Dstrbutons Let A R and let Prob(x A) denote the probablty blt that t x wll belong to A. Def. The dstrbuton functon of a random varable x s a functon defned by F(x') Prob(x x'), x' R. Key Propertes P.1 F s nondecreasng n x. P.2 lm F(x) = 1 and lm F(x) = 0. x x P.3 F s contnuous from the rght. P4 P.4 Forallx' x', Prob(x>x')=1 x') - F(x'). 2
3 Dscrete Random Varables If the random varable can assume only a fnte t number or a countable nfnte t set of values, t s sad to be a dscrete random varable. Key Propertes P.1 Prob(x = x') f(x') 0. (f s called the probablty mass functon or the probablty functon.) P.2 f( x ) Pr ob( x x ) P.3 Prob(x A) = f( x ). x A 3
4 Examples Example: #1 Consder the random varable assocated wth 2 tosses of a far con. The possble values for the #heads x are {0, 1, 2}. We have that f(0) = 1/4, f(1) = 1/2, and f(2) = 1/4. f(x) F(x) 1 X 1/2 X 3/4 X 1/4 X X 1/4 X Examples #2 A sngle toss of a far de. f(x) = 1/6 f x = 1,2,3,4,5,6. F(x ) = x /6. 4
5 Contnuous Random Varables and ther Dstrbutons Def. A random varable x has a contnuous dstrbuton f there exsts a nonnegatve functon f defned on R such that for any nterval A of R Prob (x A) = f() x dx. xa The functon f s called the probablty densty functon of x and the doman of f s called the support of the random varable x. Propertes of f P.1 f(x) 0, for all x. P.2 f ( x) dx 1. P.3 If df/dx exsts, then df/dx = f(x), for all x. In terms of geometry F(x) s the area under f(x) for x' x. 5
6 Example Example: The unform dstrbuton on [a,b]. 1/(b-a), f x [a,b] f(x) = 0, otherwse Note that F s gven by x 1 1 F(x) = [ 1 / ( b a)] dx. ( b a) x x a ( b a) ( b a) x a a Also, b b 1 a f( x) dx [ 1 / ( b a)] dx 1. ( b a) x b a b a a ( b a) ( b a) F(x) Example 1 slope =1/(b-a) -a/(b-a) a b x f(x) 1/(b-a) a b x 6
7 Dscrete Jont Dstrbutons Let the two random varables x and y have a jont probablty blt functon f(x ',, y ') = Prob(x = x ' and y = y '). Propertes of Prob Functon P.1 f(x, y ) 0. P.2 Prob((x,y ) A) = f ( x, y ). P.3 f( x, y ) = 1. ( x, y ) ( x, y ) A 7
8 The Dstrbuton Functon Defned F(x ', y ' ' ) = Prob( x x and y y ' ) = f ( x, y ), where ' L = {(x, y ) : x x and y y ' }. ( x, y ) L Margnal Prob and Dstrbuton Functons The margnal probablty functon assocated wth x s gven by f 1 (x j ) Prob(x = x j ) = f ( x, y ) y The margnal probablty functon assocated wth y s gven by f 2 (y j ) Prob(y = y j ) = f ( x, y ) x j j 8
9 Margnal dstrbuton functons The margnal dstrbuton functon of x s gven by F 1 (x j ) = Prob(x x j ) = lm yj Prob(x x j and y y j ) = lm yj F(x j,y j ). Lkewse for y, the margnal dstrbuton functon s F 2 (y j ) = lm xj F(x j,y j ). Example An example. Let x and y represent random varables representng whether or not two dfferent stocks wll ncrease or decrease n prce. Each of x and y can take on the values 0 or 1, where a 1 means that ts prce has ncreased and a 0 means that t has decreased. The probablty functon s descrbed by f(1,1) =.50 f(0,1) =.35 f(1,0) =.10 f(0,0) =.05. Answer each of the followng questons. a. Fnd F(1,0) and F(0,1). F(1,0) = =.15. F(0,1) = =.40. b. Fnd F 1 (0) = lm F(0,y) = F(0,1) =.4 y 1 c. Fnd F 2 (1) = lm F(x,1) = F(1,1) = 1. x 1 d. Fnd f 1 (0) = f( 0, y) f(0,1) + f(0,0) =.4. y e. Fnd f 1 (1) = f(, 1 y) f(1,1) +f(1,0) =.6 y 9
10 Condtonal Dstrbutons After a value of y has been observed, the probablty blt that t a value of x wll be observed s gven by Prob(x = x y = y ) = Pr ob ( x x & y y ). Pr ob( y y ) The functon g 1 (x y ) fx (, y ) f2 ( y ). s called the condtonal probablty functon of x, gven y. g 2 (y x ) s defned analogously. Propertes of Condtonal Probablty Functons () g 1 (x y ) 0. () g 1 (x y ) = x f(x,y ) / x (() and () hold for g 2 (y x )) f(x,y ) = 1. x () f(x,y ) = g 1 (x y )f 2 (y ) = g 2 (y x )f 1 (x ). 10
11 Condtonal Dstrbuton Functons F 1 (x y ) = fx (, y)/ f2 ( y), x x F 2 (y x ) = fx (, y)/ f1( x). y y The stock prce example revsted a. Compute g 1 (1 0) = f(1,0)/f 2 (0). We have that f 2 (0) = f(0,0) 0) + f(1,0) = =.15. Further f(1,0) =.1. Thus, g 1 (1 0) =.1/.15 =.66. b. Fnd g 2 (0 0) = f(0,0)/f 1 (0) =.05/.4 =.125. Here f 1 (0) = f(, 0 y ) = f(0,0) + f(0,1) = y =.4. 11
12 Contnuous Jont Dstrbutons The random varables x and y have a contnuous jont dstrbuton b t f there exsts a nonnegatve functon f defned on R 2 such that for any A R 2 Prob((x,y) A) = f( x, y) dxdy. A f s called the jont probablty densty functon of x and y. Propertes of f f satsfes the usual propertes: P.1 f(x,y) 0. P.2 f(x,y)dxdy = 1. 12
13 Dstrbuton functon F(x',y') = Prob(x x' and y y') = y' x' f(x,y)dxdy. If F s twce dfferentable, then w e have that f(x,y) = 2 F(x,y)/ x y. Margnal Densty and Dstrbuton Functons The margnal densty and dstrbuton functons are defned d as follows: a. F 1 (x) = lm y F(x,y) and F 2 (y) = lm x F(x,y). (margnal dstrbuton functons) b. f 1 (x) = fxy (,) dy and f 2 (y) = fxy (,) dx. y x 13
14 Example Let f(x,y) = 4xy for x,y [0,1] and 0 otherwse. a. Check to see that xydxdy = 1. 0 b. Fnd F(x',y'). Clearly, F(x',y') = 4 f(x,y). c. Fnd F 1 (x) and F 2 (y). We have that F 1 1( (x) = lm x 2 y 2 = x 2. y 1 Usng smlar reasonng, F 2 (y) = y 2. d. Fnd f 1 (x) and f 2 (y). y' 0 x' xydxdy = (x') 2 (y') 2. Note also that 2 F/ x y = 4xy = 0 1 f 1 (x) = f(x,y)dy = 2x and f 2 (y) = f(x,y)dx = 2y We have Condtonal Densty The condtonal densty functon of x, gven that y s fxed at a partcular value s gven by g 1 (x y) = f(x,y)/f 2 (y). Lkewse, for y we have g 2 (y x) = f(x,y)/f 1 (x). It s clear that g 1 (x y)dx = 1. 14
15 Condtonal Dstrbuton Functons We have The condtonal dstrbuton functons are gven by G 1 (x' y) = G 2 (y' x) = x' y' g 1 (x y)dx, g 2 (y x)dy. Example Let us revst example #2 above. We have that f = 4xy wth x,y (0,1). Moreover, g 1 (x y) = 4xy/2y = 2x and g 2 (y x) = 4xy/2x = 2y. x' G 1 (x' y) = 2 0 x dx = 2 ( x') 2 2 = (x') 2. By symmetry. G 2 (y' x) = (y') 2. It turns out that n ths example, x and y are ndependent random varables, because the condtonal dstrbutons do not depend on the other random varable. 15
16 Independent Random Varables Def. The random varables (x 1,...,x n ) are sad to be ndependent f for any n sets of real numbers A, we have Prob(x 1 A 1 & x 2 A 2 &...& x n A n ) = Prob(x 1 A 1 )Prob(x 2 A 2 ) Prob(x n A n ). Results on Independence The random varables x and y are ndependent d ff F(x,y) = F 1 (x)f 2 (y) or f(x,y) = f 1 (x)f 2 (y). Further, ff x and y are ndependent, then g 1 (x y) = f(x,y)/f 2 (y) = f 1 (x)f 2 (y)/ f 2 (y) = f 1 (x). 16
17 Extensons The noton of a jont dstrbuton can be extended d to any number of random varables. The margnal and condtonal dstrbutons are easly extended to ths case. Let f(x 1,...,x n ) represent the jont densty. Extensons The margnal densty for the th varable s gven by f (x ) =... f(x 1,...,x n )dx 1...dx -1 dx +1...dx n. The condtonal densty for say x 1 gven x 2,...,x n s g 1 (x 1 x 2,...,x n ) = f(x 1,...,x n )/ f(x 1,...,x n )dx 1. 17
18 Summary Measures of Probablty Dstrbutons Summary measures are scalars that convey some aspect of the dstrbuton. Because each s a scalar, all of the nformaton about the dstrbuton cannot be captured. In some cases t s of nterest to know multple summary measures of the same dstrbuton. There are two general types of measures. a. Measures of central tendency: Expectaton, t medan and mode b. measures of dsperson: Varance Expectaton The expectaton of a random varable x s gven by E(x) = x f(x ) (dscrete) E(x) = xf(x)dx. (contnuous) 18
19 Examples #1. A lottery. A church holds a lottery by sellng 1000 tckets at a dollar each. One wnner wns $750. You buy one tcket. What s your expected return? E(x) =.001(749) +.999(-1) = = The nterpretaton s that f you were to repeat ths game nfntely your long run return would be #2. You purchase 100 shares of a stock and sell them one year later. The net gan s x. The dstrbuton s gven by. (-500,.03), (-250,.07), (0,.1), (250,.25),(500,.35), (750,.15), and (1000,.05). E(x) = $ Examples #3. Let f(x) = 2x for x (0,1) and = 0, otherwse. Fnd E(x). 1 E(x) = 2x 2 dx = 2/
20 Propertes of E(x) P.1 Let g(x) be a functon of x. Then E(g(x)) s gven by E(g(x)) = g(x ) f(x ) (dscrete) E(g(x)) = g(x)f(x) dx. (contnuous) P.2 If k s a constant, then E(k) = k. P.3 Let a and b be two arbtrary constants. Then E(ax + b) = ae(x) + b. Propertes of E(x) P.4 Let x 1,...,x n be n random varables. Then E( x ) = E(x ( ). P.5 If there exsts a constant k such that Prob(x k) = 1, then E(x) k. If there exsts a constant k such that Prob(x k) = 1, then E(x) k. P.6 Let x 1,...,x n be n ndependent random varables. Then E( x ) = Ex ( ). n 1 n 1 20
21 Medan Def. If Prob(x m).5 and Prob(x m).5, then m s called a medan. a. The contnuous case m fxdx () = f() x dx =.5. m b. In the dscrete case, m need not be unque. Example: (x 1,f(x 1 )) gven by (6,.1), (8,.4), (10,.3), (15,.1), (25,.05), (50,.05). In ths case, m = 8 or 10. Mode Def. The mode s gven by m o = argmax f(x). A mode s a maxmzer of the densty functon. It need not be unque. 21
22 A Summary Measure of Dsperson: The Varance In many cases the mean the mode or the medan are not nformatve. In partcular, two dstrbutons wth the same mean can be very dfferent dstrbutons. One would lke to know how common or typcal s the mean. The varance measures ths noton by takng the expectaton of the squared devaton about the mean. Varance Def. For a random varable x, the varance s gven by E[(x- ) 2 ], where = E(x). The varance s also wrtten as Var(x) or as 2. The square root of the varance s called the standard devaton of the dstrbuton. It s wrtten as. 22
23 Illustraton Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec E(x) Var(x) Lawrence Santa Barbara Computaton: Examples a. For the dscrete case, Var(x) = (x - ) 2 f(x ). As an example, f (x, f(x )) are gven by (0,.1), (500,.8), and (1000,.1). We have that E(x) = 500. Var(x) = (0-500) 2 (.1) + ( ) 2 (.8) + ( ) 2 (.1) = 50,000. b. For the contnuous case, Var(x) = (x- ) 2 f(x)dx. Consder the example above where f = 2x wth x (0,1). From above, E(x) = 2/3. Thus, 1 Var(x) = (x - 2/3) 2 2x dx = 1/
24 Propertes of Varance P.1 Var(x) = 0 ff there exsts a c such that Prob(x = c) = 1. P.2 For any constants a and b, Var(ax +b) = a 2 V ar(x). P.3 Var(x) = E(x 2 ) - [E(x)] 2. P4 P.4 Ifx, = 1,...,n, are ndependent, then Var( x )= Var(x ). P.5 If x are ndependent, = 1,...,n, then Var( a x ) = a 2 Var(x ). A remark on moments Var (x) s sometmes called the second moment about the mean, wth E(x- ) = 0 beng the frst moment about the mean. Usng ths termnology, E(x- ) 3 s the thrd moment about the mean. It can gve us nformaton about the skewedness of the dstrbuton. E(x- ) 4 s the fourth moment about the mean and t can yeld nformaton about the modes of the dstrbuton or the peaks (kurtoss). 24
25 Moments of Condtonal and Jont Dstrbutons Gven a jont probablty densty functon f(x 1,..., x n ), the expectaton of a functon of the n varables say g(x 1,..., x n ) s defned as E(g(x 1,..., x n )) = g(x 1,..., x n ) f(x 1,..., x n ) dx 1 dx n. If the random varables are dscrete, then we would let x = (x )bethe th 1,..., x n observaton and wrte E(g(x 1,..., x n )) = g(x ) f(x ). Uncondtonal expectaton of a jont dstrbuton Gven a jont densty f(x,y), E(x) s gven by E(x) = xf 1 (x)dx = Lkewse, E(y) s xf(x,y)dxdy. E(y) = yf 2 (y)dy = yf(x,y)dxdy. 25
26 Condtonal Expectaton The condtonal expectaton of x gven that x and y are jontly dstrbuted b t d as f(x,y) s defned by (I wll gve defntons for the contnuous case only. For the dscrete case, replace ntegrals wth summatons) E(x y) = xg 1 (x y) dx Condtonal Expectaton Further the condtonal expectaton of y gven x s defned d analogously l as E(y x) = yg 2 (y x) dy 26
27 Condtonal Expectaton Note that E(E(x y)) = E(x). To see ths, compute E(E(x y)) = [ xg 1 (x y)dx]f 2 dy = { x[f(x,y)/( f 2 )]dx}f 2 dy = x f(x,y)dxd dy, and the result holds. Covarance. Covarance s a moment reflectng drecton of movement of two varables. It s defned as Cov(x,y) = E[(x- x )(y- y )]. When ths s large and postve, then x and y tend to be both much above or both much below ther respectve means at the same tme. Conversely when t s negatve. 27
28 Computaton of Cov Computaton of the covarance. Frst compute (x- x )(y- y ) = xy - x y - y x + x y. Takng E, E(xy) - x y - x y + x y = E(xy) - x y. Thus, Cov(x, y) = E(xy) - E(x)E(y). If x and y are ndependent, then E(xy) = E(x)E(y) and Cov(xy) = 0. 28
Random Variables and Probability Distributions
Random Varables and Probablty Dstrbutons Some Prelmnary Informaton Scales on Measurement IE231 - Lecture Notes 5 Mar 14, 2017 Nomnal scale: These are categorcal values that has no relatonshp of order or
More informationProblem Set 3 Solutions
Introducton to Algorthms October 4, 2002 Massachusetts Insttute of Technology 6046J/18410J Professors Erk Demane and Shaf Goldwasser Handout 14 Problem Set 3 Solutons (Exercses were not to be turned n,
More information2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements
Module 3: Element Propertes Lecture : Lagrange and Serendpty Elements 5 In last lecture note, the nterpolaton functons are derved on the bass of assumed polynomal from Pascal s trangle for the fled varable.
More informationSome Advanced SPC Tools 1. Cumulative Sum Control (Cusum) Chart For the data shown in Table 9-1, the x chart can be generated.
Some Advanced SP Tools 1. umulatve Sum ontrol (usum) hart For the data shown n Table 9-1, the x chart can be generated. However, the shft taken place at sample #21 s not apparent. 92 For ths set samples,
More informationFEATURE EXTRACTION. Dr. K.Vijayarekha. Associate Dean School of Electrical and Electronics Engineering SASTRA University, Thanjavur
FEATURE EXTRACTION Dr. K.Vjayarekha Assocate Dean School of Electrcal and Electroncs Engneerng SASTRA Unversty, Thanjavur613 41 Jont Intatve of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents
More informationFor instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)
Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A
More information6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour
6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the
More informationLOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit
LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE
More informationHermite Splines in Lie Groups as Products of Geodesics
Hermte Splnes n Le Groups as Products of Geodescs Ethan Eade Updated May 28, 2017 1 Introducton 1.1 Goal Ths document defnes a curve n the Le group G parametrzed by tme and by structural parameters n the
More informationGSLM Operations Research II Fall 13/14
GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are
More informationMonte Carlo Integration
Introducton Monte Carlo Integraton Dgtal Image Synthess Yung-Yu Chuang 11/9/005 The ntegral equatons generally don t have analytc solutons, so we must turn to numercal methods. L ( o p,ωo) = L e ( p,ωo)
More informationComplex Numbers. Now we also saw that if a and b were both positive then ab = a b. For a second let s forget that restriction and do the following.
Complex Numbers The last topc n ths secton s not really related to most of what we ve done n ths chapter, although t s somewhat related to the radcals secton as we wll see. We also won t need the materal
More informationComplex Filtering and Integration via Sampling
Overvew Complex Flterng and Integraton va Samplng Sgnal processng Sample then flter (remove alases) then resample onunform samplng: jtterng and Posson dsk Statstcs Monte Carlo ntegraton and probablty theory
More informationSolitary and Traveling Wave Solutions to a Model. of Long Range Diffusion Involving Flux with. Stability Analysis
Internatonal Mathematcal Forum, Vol. 6,, no. 7, 8 Soltary and Travelng Wave Solutons to a Model of Long Range ffuson Involvng Flux wth Stablty Analyss Manar A. Al-Qudah Math epartment, Rabgh Faculty of
More informationCompiler Design. Spring Register Allocation. Sample Exercises and Solutions. Prof. Pedro C. Diniz
Compler Desgn Sprng 2014 Regster Allocaton Sample Exercses and Solutons Prof. Pedro C. Dnz USC / Informaton Scences Insttute 4676 Admralty Way, Sute 1001 Marna del Rey, Calforna 90292 pedro@s.edu Regster
More informationElectrical analysis of light-weight, triangular weave reflector antennas
Electrcal analyss of lght-weght, trangular weave reflector antennas Knud Pontoppdan TICRA Laederstraede 34 DK-121 Copenhagen K Denmark Emal: kp@tcra.com INTRODUCTION The new lght-weght reflector antenna
More informationCHAPTER 2 DECOMPOSITION OF GRAPHS
CHAPTER DECOMPOSITION OF GRAPHS. INTRODUCTION A graph H s called a Supersubdvson of a graph G f H s obtaned from G by replacng every edge uv of G by a bpartte graph,m (m may vary for each edge by dentfyng
More informationMath Homotopy Theory Additional notes
Math 527 - Homotopy Theory Addtonal notes Martn Frankland February 4, 2013 The category Top s not Cartesan closed. problem. In these notes, we explan how to remedy that 1 Compactly generated spaces Ths
More informationMachine Learning: Algorithms and Applications
14/05/1 Machne Learnng: Algorthms and Applcatons Florano Zn Free Unversty of Bozen-Bolzano Faculty of Computer Scence Academc Year 011-01 Lecture 10: 14 May 01 Unsupervsed Learnng cont Sldes courtesy of
More informationy and the total sum of
Lnear regresson Testng for non-lnearty In analytcal chemstry, lnear regresson s commonly used n the constructon of calbraton functons requred for analytcal technques such as gas chromatography, atomc absorpton
More informationA Post Randomization Framework for Privacy-Preserving Bayesian. Network Parameter Learning
A Post Randomzaton Framework for Prvacy-Preservng Bayesan Network Parameter Learnng JIANJIE MA K.SIVAKUMAR School Electrcal Engneerng and Computer Scence, Washngton State Unversty Pullman, WA. 9964-75
More informationCS 534: Computer Vision Model Fitting
CS 534: Computer Vson Model Fttng Sprng 004 Ahmed Elgammal Dept of Computer Scence CS 534 Model Fttng - 1 Outlnes Model fttng s mportant Least-squares fttng Maxmum lkelhood estmaton MAP estmaton Robust
More informationMathematics 256 a course in differential equations for engineering students
Mathematcs 56 a course n dfferental equatons for engneerng students Chapter 5. More effcent methods of numercal soluton Euler s method s qute neffcent. Because the error s essentally proportonal to the
More informationMonte Carlo Rendering
Monte Carlo Renderng Last Tme? Modern Graphcs Hardware Cg Programmng Language Gouraud Shadng vs. Phong Normal Interpolaton Bump, Dsplacement, & Envronment Mappng Cg Examples G P R T F P D Today Does Ray
More informationModeling Local Uncertainty accounting for Uncertainty in the Data
Modelng Local Uncertanty accontng for Uncertanty n the Data Olena Babak and Clayton V Detsch Consder the problem of estmaton at an nsampled locaton sng srrondng samples The standard approach to ths problem
More informationA User Selection Method in Advertising System
Int. J. Communcatons, etwork and System Scences, 2010, 3, 54-58 do:10.4236/jcns.2010.31007 Publshed Onlne January 2010 (http://www.scrp.org/journal/jcns/). A User Selecton Method n Advertsng System Shy
More informationPolyhedral Compilation Foundations
Polyhedral Complaton Foundatons Lous-Noël Pouchet pouchet@cse.oho-state.edu Dept. of Computer Scence and Engneerng, the Oho State Unversty Feb 8, 200 888., Class # Introducton: Polyhedral Complaton Foundatons
More informationb * -Open Sets in Bispaces
Internatonal Journal of Mathematcs and Statstcs Inventon (IJMSI) E-ISSN: 2321 4767 P-ISSN: 2321-4759 wwwjmsorg Volume 4 Issue 6 August 2016 PP- 39-43 b * -Open Sets n Bspaces Amar Kumar Banerjee 1 and
More informationX- Chart Using ANOM Approach
ISSN 1684-8403 Journal of Statstcs Volume 17, 010, pp. 3-3 Abstract X- Chart Usng ANOM Approach Gullapall Chakravarth 1 and Chaluvad Venkateswara Rao Control lmts for ndvdual measurements (X) chart are
More informationNAG Fortran Library Chapter Introduction. G10 Smoothing in Statistics
Introducton G10 NAG Fortran Lbrary Chapter Introducton G10 Smoothng n Statstcs Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Smoothng Methods... 2 2.2 Smoothng Splnes and Regresson
More informationLecture #15 Lecture Notes
Lecture #15 Lecture Notes The ocean water column s very much a 3-D spatal entt and we need to represent that structure n an economcal way to deal wth t n calculatons. We wll dscuss one way to do so, emprcal
More informationLine geometry, according to the principles of Grassmann s theory of extensions. By E. Müller in Vienna.
De Lnengeometre nach den Prnzpen der Grassmanschen Ausdehnungslehre, Monastshefte f. Mathematk u. Physk, II (89), 67-90. Lne geometry, accordng to the prncples of Grassmann s theory of extensons. By E.
More informationSupport Vector Machines
/9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.
More informationAny Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation
Symmetry 2011, 3, 365-388; do:10.3390/sym3020365 OPEN ACCESS symmetry ISSN 2073-8994 www.mdp.com/journal/symmetry Artcle Any Par of 2D Curves Is Consstent wth a 3D Symmetrc Interpretaton Tadamasa Sawada
More informationData Representation in Digital Design, a Single Conversion Equation and a Formal Languages Approach
Data Representaton n Dgtal Desgn, a Sngle Converson Equaton and a Formal Languages Approach Hassan Farhat Unversty of Nebraska at Omaha Abstract- In the study of data representaton n dgtal desgn and computer
More informationUSING GRAPHING SKILLS
Name: BOLOGY: Date: _ Class: USNG GRAPHNG SKLLS NTRODUCTON: Recorded data can be plotted on a graph. A graph s a pctoral representaton of nformaton recorded n a data table. t s used to show a relatonshp
More informationAnalysis of Continuous Beams in General
Analyss of Contnuous Beams n General Contnuous beams consdered here are prsmatc, rgdly connected to each beam segment and supported at varous ponts along the beam. onts are selected at ponts of support,
More informationIntro. Iterators. 1. Access
Intro Ths mornng I d lke to talk a lttle bt about s and s. We wll start out wth smlartes and dfferences, then we wll see how to draw them n envronment dagrams, and we wll fnsh wth some examples. Happy
More informationParallelism for Nested Loops with Non-uniform and Flow Dependences
Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr
More information5 The Primal-Dual Method
5 The Prmal-Dual Method Orgnally desgned as a method for solvng lnear programs, where t reduces weghted optmzaton problems to smpler combnatoral ones, the prmal-dual method (PDM) has receved much attenton
More informationCMPS 10 Introduction to Computer Science Lecture Notes
CPS 0 Introducton to Computer Scence Lecture Notes Chapter : Algorthm Desgn How should we present algorthms? Natural languages lke Englsh, Spansh, or French whch are rch n nterpretaton and meanng are not
More informationAn Entropy-Based Approach to Integrated Information Needs Assessment
Dstrbuton Statement A: Approved for publc release; dstrbuton s unlmted. An Entropy-Based Approach to ntegrated nformaton Needs Assessment June 8, 2004 Wllam J. Farrell Lockheed Martn Advanced Technology
More informationEXTENDED BIC CRITERION FOR MODEL SELECTION
IDIAP RESEARCH REPORT EXTEDED BIC CRITERIO FOR ODEL SELECTIO Itshak Lapdot Andrew orrs IDIAP-RR-0-4 Dalle olle Insttute for Perceptual Artfcal Intellgence P.O.Box 59 artgny Valas Swtzerland phone +4 7
More informationKiran Joy, International Journal of Advanced Engineering Technology E-ISSN
Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS 0976-3945 nt Adv Engg Tech/Vol. V/ssue /Aprl-une,04/9-95 Research Paper DETERMATO O RADATVE VEW ACTOR WTOUT COSDERG TE SADOWG EECT Kran
More informationPrivate Information Retrieval (PIR)
2 Levente Buttyán Problem formulaton Alce wants to obtan nformaton from a database, but she does not want the database to learn whch nformaton she wanted e.g., Alce s an nvestor queryng a stock-market
More informationDistribution Analysis
Chapter II Dstrbuton Analyss D... (Absolute and Relatve Frequences) Let X be a characterstc possessng the attrbutesa, =,,..., k. The absolute frequency of the attrbutea, =,,..., k s defned as follows:
More informationProgramming in Fortran 90 : 2017/2018
Programmng n Fortran 90 : 2017/2018 Programmng n Fortran 90 : 2017/2018 Exercse 1 : Evaluaton of functon dependng on nput Wrte a program who evaluate the functon f (x,y) for any two user specfed values
More informationSubspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points;
Subspace clusterng Clusterng Fundamental to all clusterng technques s the choce of dstance measure between data ponts; D q ( ) ( ) 2 x x = x x, j k = 1 k jk Squared Eucldean dstance Assumpton: All features
More informationK-means and Hierarchical Clustering
Note to other teachers and users of these sldes. Andrew would be delghted f you found ths source materal useful n gvng your own lectures. Feel free to use these sldes verbatm, or to modfy them to ft your
More informationA New Approach For the Ranking of Fuzzy Sets With Different Heights
New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts Pushpnder Sngh School of Mathematcs Computer pplcatons Thapar Unversty, Patala-7 00 Inda pushpndersnl@gmalcom STCT ankng of fuzzy sets plays
More informationKent State University CS 4/ Design and Analysis of Algorithms. Dept. of Math & Computer Science LECT-16. Dynamic Programming
CS 4/560 Desgn and Analyss of Algorthms Kent State Unversty Dept. of Math & Computer Scence LECT-6 Dynamc Programmng 2 Dynamc Programmng Dynamc Programmng, lke the dvde-and-conquer method, solves problems
More informationS1 Note. Basis functions.
S1 Note. Bass functons. Contents Types of bass functons...1 The Fourer bass...2 B-splne bass...3 Power and type I error rates wth dfferent numbers of bass functons...4 Table S1. Smulaton results of type
More informationSynthesizer 1.0. User s Guide. A Varying Coefficient Meta. nalytic Tool. Z. Krizan Employing Microsoft Excel 2007
Syntheszer 1.0 A Varyng Coeffcent Meta Meta-Analytc nalytc Tool Employng Mcrosoft Excel 007.38.17.5 User s Gude Z. Krzan 009 Table of Contents 1. Introducton and Acknowledgments 3. Operatonal Functons
More informationSmoothing Spline ANOVA for variable screening
Smoothng Splne ANOVA for varable screenng a useful tool for metamodels tranng and mult-objectve optmzaton L. Rcco, E. Rgon, A. Turco Outlne RSM Introducton Possble couplng Test case MOO MOO wth Game Theory
More informationContent Based Image Retrieval Using 2-D Discrete Wavelet with Texture Feature with Different Classifiers
IOSR Journal of Electroncs and Communcaton Engneerng (IOSR-JECE) e-issn: 78-834,p- ISSN: 78-8735.Volume 9, Issue, Ver. IV (Mar - Apr. 04), PP 0-07 Content Based Image Retreval Usng -D Dscrete Wavelet wth
More informationExplicit Formulas and Efficient Algorithm for Moment Computation of Coupled RC Trees with Lumped and Distributed Elements
Explct Formulas and Effcent Algorthm for Moment Computaton of Coupled RC Trees wth Lumped and Dstrbuted Elements Qngan Yu and Ernest S.Kuh Electroncs Research Lab. Unv. of Calforna at Berkeley Berkeley
More information(1) The control processes are too complex to analyze by conventional quantitative techniques.
Chapter 0 Fuzzy Control and Fuzzy Expert Systems The fuzzy logc controller (FLC) s ntroduced n ths chapter. After ntroducng the archtecture of the FLC, we study ts components step by step and suggest a
More informationQuantifying Responsiveness of TCP Aggregates by Using Direct Sequence Spread Spectrum CDMA and Its Application in Congestion Control
Quantfyng Responsveness of TCP Aggregates by Usng Drect Sequence Spread Spectrum CDMA and Its Applcaton n Congeston Control Mehd Kalantar Department of Electrcal and Computer Engneerng Unversty of Maryland,
More informationPerformance Evaluation of Information Retrieval Systems
Why System Evaluaton? Performance Evaluaton of Informaton Retreval Systems Many sldes n ths secton are adapted from Prof. Joydeep Ghosh (UT ECE) who n turn adapted them from Prof. Dk Lee (Unv. of Scence
More informationcos(a, b) = at b a b. To get a distance measure, subtract the cosine similarity from one. dist(a, b) =1 cos(a, b)
8 Clusterng 8.1 Some Clusterng Examples Clusterng comes up n many contexts. For example, one mght want to cluster journal artcles nto clusters of artcles on related topcs. In dong ths, one frst represents
More informationRange images. Range image registration. Examples of sampling patterns. Range images and range surfaces
Range mages For many structured lght scanners, the range data forms a hghly regular pattern known as a range mage. he samplng pattern s determned by the specfc scanner. Range mage regstraton 1 Examples
More informationParameter estimation for incomplete bivariate longitudinal data in clinical trials
Parameter estmaton for ncomplete bvarate longtudnal data n clncal trals Naum M. Khutoryansky Novo Nordsk Pharmaceutcals, Inc., Prnceton, NJ ABSTRACT Bvarate models are useful when analyzng longtudnal data
More informationWishing you all a Total Quality New Year!
Total Qualty Management and Sx Sgma Post Graduate Program 214-15 Sesson 4 Vnay Kumar Kalakband Assstant Professor Operatons & Systems Area 1 Wshng you all a Total Qualty New Year! Hope you acheve Sx sgma
More informationExam 2 is Tue Nov 21. Bring a pencil and a calculator. Discuss similarity to exam1. HW3 is due Tue Dec 5.
Stat 100a: Introduction to Probability. Outline for the day 1. Bivariate and marginal density. 2. CLT. 3. CIs. 4. Sample size calculations. 5. Review for exam 2. Exam 2 is Tue Nov 21. Bring a pencil and
More informationWe are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors
We are IntechOpen, the world s leadng publsher of Open Access books Bult by scentsts, for scentsts 3,900 116,000 10M Open access books avalable Internatonal authors and edtors Downloads Our authors are
More informationALEKSANDROV URYSOHN COMPACTNESS CRITERION ON INTUITIONISTIC FUZZY S * STRUCTURE SPACE
mercan Journal of Mathematcs and cences Vol. 5, No., (January-December, 206) Copyrght Mnd Reader Publcatons IN No: 2250-302 www.journalshub.com LEKNDROV URYOHN COMPCTNE CRITERION ON INTUITIONITIC FUZZY
More informationNetwork Coding as a Dynamical System
Network Codng as a Dynamcal System Narayan B. Mandayam IEEE Dstngushed Lecture (jont work wth Dan Zhang and a Su) Department of Electrcal and Computer Engneerng Rutgers Unversty Outlne. Introducton 2.
More informationMonte Carlo 1: Integration
Monte Carlo : Integraton Prevous lecture: Analytcal llumnaton formula Ths lecture: Monte Carlo Integraton Revew random varables and probablty Samplng from dstrbutons Samplng from shapes Numercal calculaton
More informationExercises (Part 4) Introduction to R UCLA/CCPR. John Fox, February 2005
Exercses (Part 4) Introducton to R UCLA/CCPR John Fox, February 2005 1. A challengng problem: Iterated weghted least squares (IWLS) s a standard method of fttng generalzed lnear models to data. As descrbed
More informationA SYSTOLIC APPROACH TO LOOP PARTITIONING AND MAPPING INTO FIXED SIZE DISTRIBUTED MEMORY ARCHITECTURES
A SYSOLIC APPROACH O LOOP PARIIONING AND MAPPING INO FIXED SIZE DISRIBUED MEMORY ARCHIECURES Ioanns Drosts, Nektaros Kozrs, George Papakonstantnou and Panayots sanakas Natonal echncal Unversty of Athens
More informationIntroducing Environmental Variables in Nonparametric Frontier Models: a Probabilistic Approach
Journal of Productvty Analyss, 24, 93 121, 2005 2005 Sprnger Scence+Busness Meda, Inc. Manufactured n The Netherlands. Introducng Envronmental Varables n Nonparametrc Fronter Models: a Probablstc Approach
More informationSimulation: Solving Dynamic Models ABE 5646 Week 11 Chapter 2, Spring 2010
Smulaton: Solvng Dynamc Models ABE 5646 Week Chapter 2, Sprng 200 Week Descrpton Readng Materal Mar 5- Mar 9 Evaluatng [Crop] Models Comparng a model wth data - Graphcal, errors - Measures of agreement
More informationLecture 5: Multilayer Perceptrons
Lecture 5: Multlayer Perceptrons Roger Grosse 1 Introducton So far, we ve only talked about lnear models: lnear regresson and lnear bnary classfers. We noted that there are functons that can t be represented
More informationAn Optimal Algorithm for Prufer Codes *
J. Software Engneerng & Applcatons, 2009, 2: 111-115 do:10.4236/jsea.2009.22016 Publshed Onlne July 2009 (www.scrp.org/journal/jsea) An Optmal Algorthm for Prufer Codes * Xaodong Wang 1, 2, Le Wang 3,
More informationHelsinki University Of Technology, Systems Analysis Laboratory Mat Independent research projects in applied mathematics (3 cr)
Helsnk Unversty Of Technology, Systems Analyss Laboratory Mat-2.08 Independent research projects n appled mathematcs (3 cr) "! #$&% Antt Laukkanen 506 R ajlaukka@cc.hut.f 2 Introducton...3 2 Multattrbute
More informationWhy visualisation? IRDS: Visualization. Univariate data. Visualisations that we won t be interested in. Graphics provide little additional information
Why vsualsaton? IRDS: Vsualzaton Charles Sutton Unversty of Ednburgh Goal : Have a data set that I want to understand. Ths s called exploratory data analyss. Today s lecture. Goal II: Want to dsplay data
More informationComputer Animation and Visualisation. Lecture 4. Rigging / Skinning
Computer Anmaton and Vsualsaton Lecture 4. Rggng / Sknnng Taku Komura Overvew Sknnng / Rggng Background knowledge Lnear Blendng How to decde weghts? Example-based Method Anatomcal models Sknnng Assume
More informationR s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges
More informationA MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS
Proceedngs of the Wnter Smulaton Conference M E Kuhl, N M Steger, F B Armstrong, and J A Jones, eds A MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS Mark W Brantley Chun-Hung
More informationAnalysis of Collaborative Distributed Admission Control in x Networks
1 Analyss of Collaboratve Dstrbuted Admsson Control n 82.11x Networks Thnh Nguyen, Member, IEEE, Ken Nguyen, Member, IEEE, Lnha He, Member, IEEE, Abstract Wth the recent surge of wreless home networks,
More informationSpecifications in 2001
Specfcatons n 200 MISTY (updated : May 3, 2002) September 27, 200 Mtsubsh Electrc Corporaton Block Cpher Algorthm MISTY Ths document shows a complete descrpton of encrypton algorthm MISTY, whch are secret-key
More informationFusion Performance Model for Distributed Tracking and Classification
Fuson Performance Model for Dstrbuted rackng and Classfcaton K.C. Chang and Yng Song Dept. of SEOR, School of I&E George Mason Unversty FAIRFAX, VA kchang@gmu.edu Martn Lggns Verdan Systems Dvson, Inc.
More informationOn the Optimality of Spectral Compression of Meshes
On the Optmalty of Spectral Compresson of Meshes MIRELA BEN-CHEN AND CRAIG GOTSMAN Center for Graphcs and Geometrc Computng Technon Israel Insttute of Technology Spectral compresson of trangle meshes has
More informationMonte Carlo 1: Integration
Monte Carlo : Integraton Prevous lecture: Analytcal llumnaton formula Ths lecture: Monte Carlo Integraton Revew random varables and probablty Samplng from dstrbutons Samplng from shapes Numercal calculaton
More informationSteps for Computing the Dissimilarity, Entropy, Herfindahl-Hirschman and. Accessibility (Gravity with Competition) Indices
Steps for Computng the Dssmlarty, Entropy, Herfndahl-Hrschman and Accessblty (Gravty wth Competton) Indces I. Dssmlarty Index Measurement: The followng formula can be used to measure the evenness between
More informationAdaptive Transfer Learning
Adaptve Transfer Learnng Bn Cao, Snno Jaln Pan, Yu Zhang, Dt-Yan Yeung, Qang Yang Hong Kong Unversty of Scence and Technology Clear Water Bay, Kowloon, Hong Kong {caobn,snnopan,zhangyu,dyyeung,qyang}@cse.ust.hk
More informationAnalysis of Malaysian Wind Direction Data Using ORIANA
Modern Appled Scence March, 29 Analyss of Malaysan Wnd Drecton Data Usng ORIANA St Fatmah Hassan (Correspondng author) Centre for Foundaton Studes n Scence Unversty of Malaya, 63 Kuala Lumpur, Malaysa
More informationSummarizing Data using Bottom-k Sketches
Summarzng Data usng Bottom-k Sketches Edth Cohen AT&T Labs Research 8 Park Avenue Florham Park, NJ 7932, USA edth@research.att.com Ham Kaplan School of Computer Scence Tel Avv Unversty Tel Avv, Israel
More informationAssignment # 2. Farrukh Jabeen Algorithms 510 Assignment #2 Due Date: June 15, 2009.
Farrukh Jabeen Algorthms 51 Assgnment #2 Due Date: June 15, 29. Assgnment # 2 Chapter 3 Dscrete Fourer Transforms Implement the FFT for the DFT. Descrbed n sectons 3.1 and 3.2. Delverables: 1. Concse descrpton
More informationOptimal Workload-based Weighted Wavelet Synopses
Optmal Workload-based Weghted Wavelet Synopses Yoss Matas School of Computer Scence Tel Avv Unversty Tel Avv 69978, Israel matas@tau.ac.l Danel Urel School of Computer Scence Tel Avv Unversty Tel Avv 69978,
More informationBrave New World Pseudocode Reference
Brave New World Pseudocode Reference Pseudocode s a way to descrbe how to accomplsh tasks usng basc steps lke those a computer mght perform. In ths week s lab, you'll see how a form of pseudocode can be
More informationAlgorithm To Convert A Decimal To A Fraction
Algorthm To Convert A ecmal To A Fracton by John Kennedy Mathematcs epartment Santa Monca College 1900 Pco Blvd. Santa Monca, CA 90405 jrkennedy6@gmal.com Except for ths comment explanng that t s blank
More informationTaxonomy of Large Margin Principle Algorithms for Ordinal Regression Problems
Taxonomy of Large Margn Prncple Algorthms for Ordnal Regresson Problems Amnon Shashua Computer Scence Department Stanford Unversty Stanford, CA 94305 emal: shashua@cs.stanford.edu Anat Levn School of Computer
More informationAn efficient iterative source routing algorithm
An effcent teratve source routng algorthm Gang Cheng Ye Tan Nrwan Ansar Advanced Networng Lab Department of Electrcal Computer Engneerng New Jersey Insttute of Technology Newar NJ 7 {gc yt Ansar}@ntedu
More informationVanishing Hull. Jinhui Hu, Suya You, Ulrich Neumann University of Southern California {jinhuihu,suyay,
Vanshng Hull Jnhu Hu Suya You Ulrch Neumann Unversty of Southern Calforna {jnhuhusuyay uneumann}@graphcs.usc.edu Abstract Vanshng ponts are valuable n many vson tasks such as orentaton estmaton pose recovery
More informationLoop Transformations for Parallelism & Locality. Review. Scalar Expansion. Scalar Expansion: Motivation
Loop Transformatons for Parallelsm & Localty Last week Data dependences and loops Loop transformatons Parallelzaton Loop nterchange Today Scalar expanson for removng false dependences Loop nterchange Loop
More informationJoint Probabilistic Curve Clustering and Alignment
Jont Probablstc Curve Clusterng and Algnment Scott Gaffney and Padhrac Smyth School of Informaton and Computer Scence Unversty of Calforna, Irvne, CA 9697-345 {sgaffney,smyth}@cs.uc.edu Abstract Clusterng
More informationFeature Reduction and Selection
Feature Reducton and Selecton Dr. Shuang LIANG School of Software Engneerng TongJ Unversty Fall, 2012 Today s Topcs Introducton Problems of Dmensonalty Feature Reducton Statstc methods Prncpal Components
More informationMATHEMATICS FORM ONE SCHEME OF WORK 2004
MATHEMATICS FORM ONE SCHEME OF WORK 2004 WEEK TOPICS/SUBTOPICS LEARNING OBJECTIVES LEARNING OUTCOMES VALUES CREATIVE & CRITICAL THINKING 1 WHOLE NUMBER Students wll be able to: GENERICS 1 1.1 Concept of
More informationA METHOD FOR RANKING OF FUZZY NUMBERS USING NEW WEIGHTED DISTANCE
Mathematcal and omputatonal pplcatons, Vol 6, No, pp 359-369, ssocaton for Scentfc Research METHOD FOR RNKING OF FUZZY NUMERS USING NEW WEIGHTED DISTNE T llahvranloo, S bbasbandy, R Sanefard Department
More information