İstanbul Kültür University Faculty of Engineering. MCB1007 Introduction to Probability and Statistics. First Midterm. Fall
|
|
- Myrtle Bradley
- 5 years ago
- Views:
Transcription
1 İstanbul Kültü Univesity Faculty of Engineeing MCB007 Intoduction to Pobability and Statistics Fist Midtem Fall Solutions Diections You have 90 minutes to complete the exam. Please do not leave the examination oom in the fist 30 minutes of the exam. Thee ae six questions, of vaying cedit (00 points total). Indicate clealy you final answe to each question. You ae allowed to use a calculato. Duing the exam, please tun off you cell phone(s). You cannot use the book o you notes. You have one page fo cheat-sheet notes at the end of the exam papes. The answe key to this exam will be posted on Depatment of Mathematics and Compute Science boad afte the exam. Good luck! Emel Yavuz Duman, PhD. M. Fatih Uça, PhD. Question. Question 4. Question. Question 5. Question 3. Question 6. TOTAL
2 Queflion. Queflion. 5 points Thee ae n maied couples in a paty. All the paticipants shake each othe s hands only once except his/he patne. What is the total numbe of handshakes at the paty? Answe. When two people shake hands, we can think of them as foming a tempoay handshaking committee. The total numbe of handshakes will be the same as the numbe of ways of foming a committee of people fom n people (Thee ae n people at the paty since the paty consist of n maied couples). As the choices ae not odeed, we ae counting combinations; thus the total numbe of handshakes is ( ) n including patnes ones. Since all the paticipants shake each othe s hands only once except his/he patne, consideing thee ae n patnes in the paty we obtain that the total numbe of handshakes at the paty is ( ) n n n! n(n ) n n n(n ) (n )!!! 5+0points A company decided to choose 6 of its employees by dawing and give them a weekend holiday fo evey weekend duing one yea. (a) What should be the minimum numbe of employees of this company if all holiday goups ae diffeent then each othe? Answe. Let n denote the numbe of employees woking fo the company. Since thee ae 5 weekends in a yea, the inequality ( n 5 should be satisfied. Using the definition of a combination, it is easy to see that n 6. Thus fo n 6then ( 6 < 5, fo n 7then 7< 5, fo n 8then ( ) 8 6 8! 8< 5, 6!! fo n 9then ( ) 9 6 9! 84 5, 6!3! So, the minimum numbe of employees of this company is 9. (b) It is given that the numbe of the employees of this company is equal to the minimum numbe that you find in pat (a). Also we know that two bothes ae woking fo this company. What is the pobability of selecting thei names consecutively in the fist dawing? Answe. Let we define an event A {Bothe s names ae selected consecutively in the fist dawing}. Since the numbe of employees woking fo the company is 9, the pobability of selecting thei names consecutively in the fist dawing is P (A) ( ) ) 4 5!! 9P 6 7! 5!! 4!3! 5 9! 36. 3! MCB007 - Int. to Pob. and Statistics Fist Midtem
3 Queflion 3. ( (a) Find the coefficient of in the expansion of x 4 Answe. Using the Binomial coefficient, we obtain x + 3 x ) points ( x + 3 x ) 7 (x ) / 7 ( ) x /3 x ( 7)/ x /3 ( ) x x + 6. Since x + 6 x 4 + 4, 6 thus we see that 3. So, the coefficient of x 4 is ! 3!4! 35. (b) In a goup of 6 maied couple, 4 people ae selected at andom. What is the pobability that NOT maied couple is selected? Answe. Let we define an event A {only one peson fom a couple is selected}. So, the pobability that not maied couple selected is 0 P (A) ( 6 ) 4 4 ( ) MCB007 - Int. to Pob. and Statistics 3 Fist Midtem
4 Queflion 4. 5 points Show that if events A and B ae independent then events A and B ae independent. Answe. Since A and B ae independent events then we know that A and B ae also independent. So, P (A B )P(A)P(B ). On the othe hand, it is easy to see that the B (A B ) (A B ). Since A B and A B ae mutually exclusive, and A and B ae independent by the assumption, we have It follows that P (B )P[(A B ) (A B )] P (A B )+P(A B ) (by Postulate 3) P (A)P (B )+P(A B ). hence that A and B ae independent. P (A B )P(B ) P (A)P (B ) P (B )[ P (A)] P (B )P (A ) Queflion points A continuous andom vaiable X has the following pobability density function { kx 4, x >, f(x) 0, elsewhee. (a) Find k. Answe. f(x)dx f(x)dx + f(x)dx lim kx 4 dx lim k x 3 c c 3 k 3 lim c (c 3 0 ) k 3 (0 ) k 3 k 3. c c (b) Find the distibution function of the andom vaiable X. Answe. If x then F (x) x f(u)du 0 If x> then F (x) x f(u)du f(u)du + x f(u)du x 3u 4 du 3u 3 3 x 3 F (x) x { 0, x,, x 3 x >. MCB007 - Int. to Pob. and Statistics 4 Fist Midtem
5 Queflion points Suppose that 3 calculatos ae andomly chosen without eplacement fom the following goup of 0 calculatos: 7 new, used (woking) and out of ode (not woking). Let X denotes the numbe of new calculatos chosen and Y denotes the numbe of used calculatos chosen. (a) Find the joint pobability distibution table. Answe. Though X can take on values 0,, and 3, andy cantakeonvalues0 and, when we conside them jointly, X + Y 3. So, not all combinations of (X, Y ) ae possible. Since thee ae ( ) 0 3 diffeent ways to choose 3 out of 0, then ( )( f(0, ) ),f(, 0) )( f(, 0) ) 4,f(, ) )( ) 7,f(, ) )( ),f(3, 0) )( )( ) 3) Theefoe, we obtain the joint pobability distibution P (X x, Y y) f(x, y) fo (X, Y ): y x 0 g(x) 0 / / 7/ 4/ / 4/ / 63/ 3 35/ 35/ h(y) 84/ 36/ (b) Find the conditional distibution of Y given X. Answe. Since the conditional distibution of Y given X is given by w(y ) f(,y) g() f(,y) 63/, then w(0 ) f(, 0) 63/ 4/ f(, ) 4/63, w( ) 63/ 63/ / 63/ /63. (c) Detemine whethe o not X and Y ae independent. Answe. If X and Y ae independent then f(x, y) g(x)h(y) fo all x 0,,, 3 and y 0,. Let we conside (x, y) (0, 0). Since f(0, 0) 0 83 g(0) h(0), we see that X and Y ae dependent. MCB007 - Int. to Pob. and Statistics 5 Fist Midtem
Question 1. Question 4. Question 2. Question 5. Question 3. Question 6.
İstanbul Kültür University Faculty of Engineering MCB17 Introduction to Probability Statistics Second Midterm Fall 21-21 Number: Name: Department: Section: Directions You have 9 minutes to complete the
More informationMCB1007 Introduction to Probability and Statistics. First Midterm. Fall Solutions
İstanbul Kültür University MCB7 Introduction to Probability and Statistics First Midterm Fall 4-5 Solutions Directions You have 9 minutes to complete the eam Please do not leave the eamination room in
More information1) (A B) = A B ( ) 2) A B = A. i) A A = φ i j. ii) Additional Important Properties of Sets. De Morgan s Theorems :
Additional Impotant Popeties of Sets De Mogan s Theoems : A A S S Φ, Φ S _ ( A ) A ) (A B) A B ( ) 2) A B A B Cadinality of A, A, is defined as the numbe of elements in the set A. {a,b,c} 3, { }, while
More informationMath 151. Rumbos Spring Solutions to Assignment #7
Math. Rumbos Sping 202 Solutions to Assignment #7. Fo each of the following, find the value of the constant c fo which the given function, p(x, is the pobability mass function (pmf of some discete andom
More informationRandom Variables and Probability Distribution Random Variable
Random Vaiables and Pobability Distibution Random Vaiable Random vaiable: If S is the sample space P(S) is the powe set of the sample space, P is the pobability of the function then (S, P(S), P) is called
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More informationand the correct answer is D.
@. Assume the pobability of a boy being bon is the same as a gil. The pobability that in a family of 5 childen thee o moe childen will be gils is given by A) B) C) D) Solution: The pobability of a gil
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationEN40: Dynamics and Vibrations. Midterm Examination Thursday March
EN40: Dynamics and Vibations Midtem Examination Thusday Mach 9 2017 School of Engineeing Bown Univesity NAME: Geneal Instuctions No collaboation of any kind is pemitted on this examination. You may bing
More informationSection 5.3 Arrangements and Selections with repetitions
Section 5.3 Aangements and Selections with epetitions Example 1: The numbe of aangements of BANANA? Thm 1: Given n objects, 1 of type 1, 2 of type 2,..., m of type m, with n = 1 + 2 + m, then the numbe
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationMotithang Higher Secondary School Thimphu Thromde Mid Term Examination 2016 Subject: Mathematics Full Marks: 100
Motithang Highe Seconday School Thimphu Thomde Mid Tem Examination 016 Subject: Mathematics Full Maks: 100 Class: IX Witing Time: 3 Hous Read the following instuctions caefully In this pape, thee ae thee
More informationEXTRA HOTS PROBLEMS. (5 marks) Given : t 3. = a + (n 1)d = 3p 2q + (n 1) (q p) t 10. = 3p 2q + (10 1) (q p) = 3p 2q + 9 (q p) = 3p 2q + 9q 9p = 7q 6p
MT EDUCARE LTD. EXTRA HOTS PROBLEMS HOTS SUMS CHAPTER : - ARITHMETIC PROGRESSION AND GEOMETRIC PROGRESSION. If 3 d tem of an A.P. is p and the 4 th tem is q. Find its n th tem and hence find its 0 th tem.
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationTHE NUMBER OF TWO CONSECUTIVE SUCCESSES IN A HOPPE-PÓLYA URN
TH NUMBR OF TWO CONSCUTIV SUCCSSS IN A HOPP-PÓLYA URN LARS HOLST Depatment of Mathematics, Royal Institute of Technology S 100 44 Stocholm, Sweden -mail: lholst@math.th.se Novembe 27, 2007 Abstact In a
More informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R Pena Towe, Road No, Contactos Aea, Bistupu, Jamshedpu 8, Tel (657)89, www.penaclasses.com IIT JEE Mathematics Pape II PART III MATHEMATICS SECTION I Single Coect Answe Type This section contains 8 multiple
More informationAMC 10 Contest B. Solutions Pamphlet. Wednesday, FEBRUARY 21, American Mathematics Competitions
The MATHEMATICAL ASSOCIATION of AMERICA Ameican Mathematics Competitions 8 th Annual Ameican Mathematics Contest 10 AMC 10 Contest B Solutions Pamphlet Wednesday, FEBRUARY 21, 2007 This Pamphlet gives
More information1. Review of Probability.
1. Review of Pobability. What is pobability? Pefom an expeiment. The esult is not pedictable. One of finitely many possibilities R 1, R 2,, R k can occu. Some ae pehaps moe likely than othes. We assign
More informationST 501 Course: Fundamentals of Statistical Inference I. Sujit K. Ghosh.
ST 501 Couse: Fundamentals of Statistical Infeence I Sujit K. Ghosh sujit.ghosh@ncsu.edu Pesented at: 2229 SAS Hall, Depatment of Statistics, NC State Univesity http://www.stat.ncsu.edu/people/ghosh/couses/st501/
More informationHOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS?
6th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE HOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS? Cecília Sitkuné Göömbei College of Nyíegyháza Hungay Abstact: The
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More informationCentral Coverage Bayes Prediction Intervals for the Generalized Pareto Distribution
Statistics Reseach Lettes Vol. Iss., Novembe Cental Coveage Bayes Pediction Intevals fo the Genealized Paeto Distibution Gyan Pakash Depatment of Community Medicine S. N. Medical College, Aga, U. P., India
More informationFW Laboratory Exercise. Survival Estimation from Banded/Tagged Animals. Year No. i Tagged
FW66 -- Laboatoy Execise uvival Estimation fom Banded/Tagged Animals Conside a geogaphically closed population of tout (Youngs and Robson 97). The adults ae tagged duing fall spawning, and subsequently
More information1 Notes on Order Statistics
1 Notes on Ode Statistics Fo X a andom vecto in R n with distibution F, and π S n, define X π by and F π by X π (X π(1),..., X π(n) ) F π (x 1,..., x n ) F (x π 1 (1),..., x π 1 (n)); then the distibution
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationEXAM NMR (8N090) November , am
EXA NR (8N9) Novembe 5 9, 9. 1. am Remaks: 1. The exam consists of 8 questions, each with 3 pats.. Each question yields the same amount of points. 3. You ae allowed to use the fomula sheet which has been
More informationK.S.E.E.B., Malleshwaram, Bangalore SSLC Model Question Paper-1 (2015) Mathematics
K.S.E.E.B., Malleshwaam, Bangaloe SSLC Model Question Pape-1 (015) Mathematics Max Maks: 80 No. of Questions: 40 Time: Hous 45 minutes Code No. : 81E Fou altenatives ae given fo the each question. Choose
More informationPermutations and Combinations
Pemutations and Combinations Mach 11, 2005 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication Pinciple
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationPHYS 1410, 11 Nov 2015, 12:30pm.
PHYS 40, Nov 205, 2:30pm. A B = AB cos φ x = x 0 + v x0 t + a 2 xt 2 a ad = v2 2 m(v2 2 v) 2 θ = θ 0 + ω 0 t + 2 αt2 L = p fs µ s n 0 + αt K = 2 Iω2 cm = m +m 2 2 +... m +m 2 +... p = m v and L = I ω ω
More informationReview for the previous lecture
Review fo the pevious letue Definition: sample spae, event, opeations (union, intesetion, omplementay), disjoint, paiwise disjoint Theoem: ommutatitivity, assoiativity, distibution law, DeMogan s law Pobability
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationAP Physics C: Electricity and Magnetism 2001 Scoring Guidelines
AP Physics C: Electicity and Magnetism 1 Scoing Guidelines The mateials included in these files ae intended fo non-commecial use by AP teaches fo couse and exam pepaation; pemission fo any othe use must
More informationIntroduction and Vectors
SOLUTIONS TO PROBLEMS Intoduction and Vectos Section 1.1 Standads of Length, Mass, and Time *P1.4 Fo eithe sphee the volume is V = 4! and the mass is m =!V =! 4. We divide this equation fo the lage sphee
More informationA Comparison and Contrast of Some Methods for Sample Quartiles
A Compaison and Contast of Some Methods fo Sample Quatiles Anwa H. Joade and aja M. Latif King Fahd Univesity of Petoleum & Mineals ABSTACT A emainde epesentation of the sample size n = 4m ( =, 1, 2, 3)
More informationLab #4: Newton s Second Law
Lab #4: Newton s Second Law Si Isaac Newton Reading Assignment: bon: Januay 4, 1643 Chapte 5 died: Mach 31, 1727 Chapte 9, Section 9-7 Intoduction: Potait of Isaac Newton by Si Godfey Knelle http://www.newton.cam.ac.uk/at/potait.html
More informationPledge: Signature:
S 202, Sing 2005 Midtem 1: 24 eb 2005 Page 1/8 Name: KEY E-mail D: @viginia.edu Pledge: Signatue: Thee ae 75 minutes fo this exam and 100 oints on the test; don t send too long on any one uestion! The
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationON THE RECURRENCE OF RANDOM WALKS
ON THE RECURRENCE OF RANDOM WALKS SIMON LAZARUS Abstact. We define andom walks on and ecuent points and demonstate that a andom walk s ecuence to implies its ecuence to each of its possible points. We
More informationHistory of Astronomy - Part II. Tycho Brahe - An Observer. Johannes Kepler - A Theorist
Histoy of Astonomy - Pat II Afte the Copenican Revolution, astonomes stived fo moe obsevations to help bette explain the univese aound them Duing this time (600-750) many majo advances in science and astonomy
More informationExploration of the three-person duel
Exploation of the thee-peson duel Andy Paish 15 August 2006 1 The duel Pictue a duel: two shootes facing one anothe, taking tuns fiing at one anothe, each with a fixed pobability of hitting his opponent.
More information3.6 Applied Optimization
.6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the
More informationOnline Mathematics Competition Wednesday, November 30, 2016
Math@Mac Online Mathematics Competition Wednesday, Novembe 0, 206 SOLUTIONS. Suppose that a bag contains the nine lettes of the wod OXOMOXO. If you take one lette out of the bag at a time and line them
More informationVoltage ( = Electric Potential )
V-1 of 10 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More informationMultiple Experts with Binary Features
Multiple Expets with Binay Featues Ye Jin & Lingen Zhang Decembe 9, 2010 1 Intoduction Ou intuition fo the poect comes fom the pape Supevised Leaning fom Multiple Expets: Whom to tust when eveyone lies
More informationPHYSICS 1210 Exam 2 University of Wyoming 14 March ( Day!) points
PHYSICS 1210 Exam 2 Univesity of Wyoming 14 Mach ( Day!) 2013 150 points This test is open-note and closed-book. Calculatos ae pemitted but computes ae not. No collaboation, consultation, o communication
More informationPhysics 211: Newton s Second Law
Physics 211: Newton s Second Law Reading Assignment: Chapte 5, Sections 5-9 Chapte 6, Section 2-3 Si Isaac Newton Bon: Januay 4, 1643 Died: Mach 31, 1727 Intoduction: Kinematics is the study of how objects
More informationMATHEMATICS GRADE 12 SESSION 38 (LEARNER NOTES)
EXAM PREPARATION PAPER (A) Leane Note: In this session you will be given the oppotunity to wok on a past examination pape (Feb/Ma 010 DoE Pape ). The pape consists of 1 questions. Question 1, and will
More informationBetween any two masses, there exists a mutual attractive force.
YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationQIP Course 10: Quantum Factorization Algorithm (Part 3)
QIP Couse 10: Quantum Factoization Algoithm (Pat 3 Ryutaoh Matsumoto Nagoya Univesity, Japan Send you comments to yutaoh.matsumoto@nagoya-u.jp Septembe 2018 @ Tokyo Tech. Matsumoto (Nagoya U. QIP Couse
More informationReliability analysis examples
Reliability analysis examples Engineeing Risk Analysis Goup, Technische Univesität München. Acisst., 80 Munich, Gemany. May 8, 08 Intoduction In the context of eliability analysis and ae event estimation,
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More information763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012
763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012 1. Continuous Random Walk Conside a continuous one-dimensional andom walk. Let w(s i ds i be the pobability that the length of the i th displacement
More informationPhysics 121 Hour Exam #5 Solution
Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given
More information18.06 Problem Set 4 Solution
8.6 Poblem Set 4 Solution Total: points Section 3.5. Poblem 2: (Recommended) Find the lagest possible numbe of independent vectos among ) ) ) v = v 4 = v 5 = v 6 = v 2 = v 3 =. Solution (4 points): Since
More informationLab #0. Tutorial Exercises on Work and Fields
Lab #0 Tutoial Execises on Wok and Fields This is not a typical lab, and no pe-lab o lab epot is equied. The following execises will emind you about the concept of wok (fom 1130 o anothe intoductoy mechanics
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationReview Exercise Set 16
Review Execise Set 16 Execise 1: A ectangula plot of famland will be bounded on one side by a ive and on the othe thee sides by a fence. If the fame only has 600 feet of fence, what is the lagest aea that
More information16 Modeling a Language by a Markov Process
K. Pommeening, Language Statistics 80 16 Modeling a Language by a Makov Pocess Fo deiving theoetical esults a common model of language is the intepetation of texts as esults of Makov pocesses. This model
More information1. Show that the volume of the solid shown can be represented by the polynomial 6x x.
7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationAppendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk
Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating
More informationMEASURING CHINESE RISK AVERSION
MEASURING CHINESE RISK AVERSION --Based on Insuance Data Li Diao (Cental Univesity of Finance and Economics) Hua Chen (Cental Univesity of Finance and Economics) Jingzhen Liu (Cental Univesity of Finance
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationWeb-based Supplementary Materials for. Controlling False Discoveries in Multidimensional Directional Decisions, with
Web-based Supplementay Mateials fo Contolling False Discoveies in Multidimensional Diectional Decisions, with Applications to Gene Expession Data on Odeed Categoies Wenge Guo Biostatistics Banch, National
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationGaia s Place in Space
Gaia s Place in Space The impotance of obital positions fo satellites Obits and Lagange Points Satellites can be launched into a numbe of diffeent obits depending on thei objectives and what they ae obseving.
More informationPHYSICS W term 2
PHYSICS 153 08W tem Electicity, Magnetism, Electomagnetic Waves, Optics Pof. W. McCutcheon Henn. 81 604-8-634 mccutche@phas.ubc.ca Office hous: Monday 10:30-11:30 Fiday 10:30-11:30 o by appointment Text:
More informationFailure Probability of 2-within-Consecutive-(2, 2)-out-of-(n, m): F System for Special Values of m
Jounal of Mathematics and Statistics 5 (): 0-4, 009 ISSN 549-3644 009 Science Publications Failue Pobability of -within-consecutive-(, )-out-of-(n, m): F System fo Special Values of m E.M.E.. Sayed Depatment
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More informationMultiple Criteria Secretary Problem: A New Approach
J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and
More informationIntroduction to Mathematical Statistics Robert V. Hogg Joeseph McKean Allen T. Craig Seventh Edition
Intoduction to Mathematical Statistics Robet V. Hogg Joeseph McKean Allen T. Caig Seventh Edition Peason Education Limited Edinbugh Gate Halow Essex CM2 2JE England and Associated Companies thoughout the
More informationPhysics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism
Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up
More informationAP Centripetal Acceleration Lab
AP PHYSICS NAME: PERIOD: DATE: GRADE: DEVIL PHYSICS BADDEST CLASS ON CAMPUS AP Centipetal Acceleation Lab Note: Data collection will be done by table goups. Data analysis is to be done individually. Copying
More informationOn the Poisson Approximation to the Negative Hypergeometric Distribution
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution
More informationCircular motion. Objectives. Physics terms. Assessment. Equations 5/22/14. Describe the accelerated motion of objects moving in circles.
Cicula motion Objectives Descibe the acceleated motion of objects moving in cicles. Use equations to analyze the acceleated motion of objects moving in cicles.. Descibe in you own wods what this equation
More informationSOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS
Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal
More informationAP Physics C: Electricity and Magnetism 2003 Scoring Guidelines
AP Physics C: Electicity and Magnetism 3 Scoing Guidelines The mateials included in these files ae intended fo use by AP teaches fo couse and exam pepaation; pemission fo any othe use must be sought fom
More informationSAMPLE QUIZ 3 - PHYSICS For a right triangle: sin θ = a c, cos θ = b c, tan θ = a b,
SAMPLE QUIZ 3 - PHYSICS 1301.1 his is a closed book, closed notes quiz. Calculatos ae pemitted. he ONLY fomulas that may be used ae those given below. Define all symbols and justify all mathematical expessions
More informationGCSE MATHEMATICS FORMULAE SHEET HIGHER TIER
Pythagoas Volume of cone = Theoem c a a + b = c hyp coss section adj b opp length Intenational GCSE MATHEMATICS FORMULAE SHEET HIGHER TIER Cuved suface aea of cone = adj = hyp opp = hyp opp = adj o sin
More informationn 1 Cov(X,Y)= ( X i- X )( Y i-y ). N-1 i=1 * If variable X and variable Y tend to increase together, then c(x,y) > 0
Covaiance and Peason Coelation Vatanian, SW 540 Both covaiance and coelation indicate the elationship between two (o moe) vaiables. Neithe the covaiance o coelation give the slope between the X and Y vaiable,
More informationMATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form
MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and
More informationMovie Review Part One due Tuesday (in class) please print
Movie Review Pat One due Tuesday (in class) please pint Test in class on Fiday. You may stat at 8:30 if you want. (The topic of powe is not on test.) Chaptes 4-6 Main Ideas in Class Today Afte class, you
More informationarxiv: v2 [physics.data-an] 15 Jul 2015
Limitation of the Least Squae Method in the Evaluation of Dimension of Factal Bownian Motions BINGQIANG QIAO,, SIMING LIU, OUDUN ZENG, XIANG LI, and BENZONG DAI Depatment of Physics, Yunnan Univesity,
More informationA Tutorial on Multiple Integrals (for Natural Sciences / Computer Sciences Tripos Part IA Maths)
A Tutoial on Multiple Integals (fo Natual Sciences / Compute Sciences Tipos Pat IA Maths) Coections to D Ian Rud (http://people.ds.cam.ac.uk/ia/contact.html) please. This tutoial gives some bief eamples
More informationAPPLICATION OF MAC IN THE FREQUENCY DOMAIN
PPLICION OF MC IN HE FREQUENCY DOMIN D. Fotsch and D. J. Ewins Dynamics Section, Mechanical Engineeing Depatment Impeial College of Science, echnology and Medicine London SW7 2B, United Kingdom BSRC he
More information6 Matrix Concentration Bounds
6 Matix Concentation Bounds Concentation bounds ae inequalities that bound pobabilities of deviations by a andom vaiable fom some value, often its mean. Infomally, they show the pobability that a andom
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationAlgebra. Substitution in algebra. 3 Find the value of the following expressions if u = 4, k = 7 and t = 9.
lgeba Substitution in algeba Remembe... In an algebaic expession, lettes ae used as substitutes fo numbes. Example Find the value of the following expessions if s =. a) s + + = = s + + = = Example Find
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationCircular Orbits. and g =
using analyse planetay and satellite motion modelled as unifom cicula motion in a univesal gavitation field, a = v = 4π and g = T GM1 GM and F = 1M SATELLITES IN OBIT A satellite is any object that is
More informationPES 3950/PHYS 6950: Homework Assignment 6
PES 3950/PHYS 6950: Homewok Assignment 6 Handed out: Monday Apil 7 Due in: Wednesday May 6, at the stat of class at 3:05 pm shap Show all woking and easoning to eceive full points. Question 1 [5 points]
More informationPhysics 1C Fall 2011: Quiz 1 Version A 1
Physics 1C Fall 2011: Quiz 1 Vesion A 1 Depatment of Physics Physics 1C Fall Quate - 2011 D. Mak Paddock INSTRUCTIONS: 1. Pint you full name below LAST NAME FIRST NAME MIDDLE INITIAL 2. You code numbe
More informationTrigonometry Standard Position and Radians
MHF 4UI Unit 6 Day 1 Tigonomety Standad Position and Radians A. Standad Position of an Angle teminal am initial am Angle is in standad position when the initial am is the positive x-axis and the vetex
More informationMath 1105: Calculus I (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 3
Math : alculus I Math/Sci majos MWF am / pm, ampion Witten homewok Review: p 94, p 977,8,9,6, 6: p 46, 6: p 4964b,c,69, 6: p 47,6 p 94, Evaluate the following it by identifying the integal that it epesents:
More information