Week 03 Discussion. 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 2% die.
|
|
- Edith Simon
- 5 years ago
- Views:
Transcription
1 STAT 400 Wee 03 Discussio Fall 07. ~.5- ~.6- At the begiig of a cetai study of a gou of esos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the five-yea study, it was detemied that the death ates of the heavy smoes ad light smoes wee seve ad thee times that of the osmoes, esectively. A adomly selected aticiat died ove the five-yea eiod. What is the obability that the aticiat was a osmoe? A light smoe? A heavy smoe?. At a hosital s emegecy oom, atiets ae classified ad 0% of them ae citical, 30% ae seious, ad 50% ae stable. Of the citical oes, 30% die; of the seious, 0% die; ad of the stable, % die. a) ~.5-5 ~.6-5 ~ Give that a atiet dies, what is the coditioal obability that the atiet was classified as citical? As seious? As stable? b) Ae evets {a atiet dies} ad {a atiet was classified as citical} ideedet? Justify you aswe. c) Ae evets {a atiet dies} ad {a atiet was classified as seious} ideedet? Justify you aswe. 3. Alex leas that his favoite socce team, Ubaa-Chamaig Uited ( ), has a 70% chace of sigig oe of the best layes i the wold, Ro Aldo. He immediately us some comute simulatios ad discoves that if sigs Ro Aldo, it would have a 0.90 obability of wiig the Ameica Cetal Illiois Divisio chamioshi. Ufotuately, if does ot sig Ro Aldo, the the obability of wiig the chamioshi is oly Alex becomes too excited ad slis ito a coma. He comes out of the coma a yea late ad fids out that has wo the chamioshi. What is the obability that was able to sig Ro Aldo?
2 4. Jac, Mie ad Tom ae oommates, ad evey Suday ight they slit a lage izza fo die. Whe thee is oly oe slice left, the obability that Jac wats it is 0.40, the obability that Mie wats it is 0.35, ad the obability that Tom wats it is 0.5. Suose that whethe o ot each oe of them will wat the last slice is ideedet of the othe two. a) What is the obability that oly oe of the oommates will wat the last slice? b) What is the obability that at least oe of the oommates will wat the last slice? c) What is the obability that at most oe of the oommates will wat the last slice? 5. Dug A is effective with obability Dug B is effective with obability Thee is a 40% chace of a egative dug iteactio betwee dugs A ad B. Suose the effectiveess of the two dugs ad the ossibility of a egative dug iteactio ae all ideedet. Fid the obability that a) both dugs ae effective ad thee is o egative dug iteactio. b) at least oe of the two dugs is effective ad thee is o egative dug iteactio. c) at most oe of the two dugs is effective ad thee is egative dug iteactio. 6. A ba has two emegecy souces of owe fo its comutes. Thee is a 95% chace that souce will oeate duig a total owe failue, ad a 80% chace that souce will oeate. Assume the owe souces ae ideedet. What is the obability that at least oe of them will oeate duig a total owe failue? If P ( A ) 0.3, P ( B ) 0.6. a) Fid P ( A B ) whe A ad B ae ideedet. b) Fid P ( A B ) whe A ad B ae mutually exclusive.
3 If P ( A ) 0.8, P ( B ) 0.5, ad P ( A B ) 0.9, ae A ad B ideedet evets? Why o why ot? Die A has oage o oe face ad blue o five faces, Die B has oage o two faces ad blue o fou faces, Die C has oage o thee faces ad blue o thee faces. All ae fai dice. If the thee dice ae olled, fid the obability that exactly two of the thee dice come u oage? 0. A electoic device has fou ideedet comoets. Two of those fou ae ew, ad have a eliability of 0.80 each, oe is old, with 0.75 eliability, ad oe is vey old, ad its eliability is a) Suose that the device wos if all fou comoets ae fuctioal. What is the obability that the device will wo whe eeded? b) Suose that the device wos if at least oe of the fou comoets is fuctioal. What is the obability that the device will wo whe eeded? c) Suose that the fou comoets ae coected as show o the diagam below. Fid the eliability of the system A oal fial exam cotiues util a studet eithe aswes two questios i a ow coectly (ad asses) o aswes two questios i a ow icoectly (ad fails). Suose Alex has obability to aswe ay questio coectly, ideedetly of ay othe questios. What is the obability that Alex would ass the exam?
4 . Suose Jae has a fai 4-sided die, ad Dic has a fai 6-sided die. Each day, they oll thei dice at the same time (ideedetly) util someoe olls a (as may times as ecessay). (The the eso who did ot oll a does the dishes.) Fid the obability that Jae olls the fist befoe Dic does. 3. Pove (show) that a) b) ( Pascal s equatio ).. 4. We aleady ow that 0. Pove (show) that The eatig club is hostig a mae-you-ow sudae aty at which the followig ae ovided: Ice Ceam Flavos Chocolate Cooies ceam Stawbey Vailla Toigs Caamel Hot fudge Mashmallow M&M s Nuts Stawbeies a) How may sudaes ae ossible usig oe flavo of ice ceam ad thee diffeet toigs? b) How may sudaes ae ossible usig oe flavo of ice ceam ad fom zeo to six toigs? c) How may diffeet combiatios of flavos of thee scoos of ice ceam ae ossible if it is emissible to mae all thee scoos the same flavo?
5 Aswes:. ~.5- ~.6- At the begiig of a cetai study of a gou of esos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the five-yea study, it was detemied that the death ates of the heavy smoes ad light smoes wee seve ad thee times that of the osmoes, esectively. A adomly selected aticiat died ove the five-yea eiod. What is the obability that the aticiat was a osmoe? A light smoe? A heavy smoe? P ( H ) 0.5, P ( H ) 0.30, P ( N ) 0.55, P ( D H ) 7, P ( D L ) 3, P ( D N ). P ( D ) P ( H ) P ( D H ) + P ( L ) P ( D L ) + P ( N ) P ( D N ) P ( H D ) P ( L D ) P ( N D )
6 . At a hosital s emegecy oom, atiets ae classified ad 0% of them ae citical, 30% ae seious, ad 50% ae stable. Of the citical oes, 30% die; of the seious, 0% die; ad of the stable, % die. a) ~.5-5 ~.6-5 ~ Give that a atiet dies, what is the coditioal obability that the atiet was classified as citical? As seious? As stable? P ( atiet dies ) P ( citical atiet dies ) P ( seious atiet dies ) P ( stable atiet dies ) b) Ae evets {a atiet dies} ad {a atiet was classified as citical} ideedet? Justify you aswe. P ( atiet dies citical ) P ( atiet dies ) P ( citical ). P ( atiet dies citical ) P ( atiet dies ). P ( citical atiet dies ) P ( citical ). {a atiet dies} ad {a atiet was classified as citical} ae NOT ideedet. c) Ae evets {a atiet dies} ad {a atiet was classified as seious} ideedet? Justify you aswe. P ( atiet dies seious ) P ( atiet dies ) P ( seious ). P ( atiet dies seious ) P ( atiet dies ). P ( seious atiet dies ) P ( seious ). {a atiet dies} ad {a atiet was classified as citical} ae ideedet.
7 3. Alex leas that his favoite socce team, Ubaa-Chamaig Uited ( ), has a 70% chace of sigig oe of the best layes i the wold, Ro Aldo. He immediately us some comute simulatios ad discoves that if sigs Ro Aldo, it would have a 0.90 obability of wiig the Ameica Cetal Illiois Divisio chamioshi. Ufotuately, if does ot sig Ro Aldo, the the obability of wiig the chamioshi is oly Alex becomes too excited ad slis ito a coma. He comes out of the coma a yea late ad fids out that has wo the chamioshi. What is the obability that was able to sig Ro Aldo? P ( RA ) 0.70, P ( W RA ) 0.90, P ( W RA' ) Bayes Theoem: P ( RA W ) P( RA ) P( W RA ) P( RA ) P( W RA ) P( RA' ) P( W RA' ) OR W W' RA RA' P ( RA W )
8 4. Jac, Mie ad Tom ae oommates, ad evey Suday ight they slit a lage izza fo die. Whe thee is oly oe slice left, the obability that Jac wats it is 0.40, the obability that Mie wats it is 0.35, ad the obability that Tom wats it is 0.5. Suose that whethe o ot each oe of them will wat the last slice is ideedet of the othe two. ( Jac ) 0.40, P( Jac' ) 0.60, ( Mie ) 0.35, P( Mie' ) 0.65, ( Tom ) 0.5, P( Tom' ) a) What is the obability that oly oe of the oommates will wat the last slice? oly Jac Jac Mie' Tom' , oly Mie Jac' Mie Tom' , oly Tom Jac' Mie' Tom P( oly oe wats the last slice ) P( oly Jac o oly Mie o oly Tom ) P( oly Jac ) + P( oly Mie ) + P( oly Tom ) b) What is the obability that at least oe of the oommates will wat the last slice? P( at least oe wats the last slice ) P( o oe wats the last slice ) P( Jac' Mie' Tom' ) c) What is the obability that at most oe of the oommates will wat the last slice? P( at most oe wats the last slice ) P( oly oe wats the last slice ) + P( o oe wats the last slice )
9 5. Dug A is effective with obability Dug B is effective with obability Thee is a 40% chace of a egative dug iteactio betwee dugs A ad B. Suose the effectiveess of the two dugs ad the ossibility of a egative dug iteactio ae all ideedet. Fid the obability that a) both dugs ae effective ad thee is o egative dug iteactio. ( Dug A is effective ) AND ( Dug B is effective ) AND ( o egative dug iteactio ) b) at least oe of the two dugs is effective ad thee is o egative dug iteactio. [ ] OR [ ( 0.80 ) ( 0.70 ) ] OR c) at most oe of the two dugs is effective ad thee is egative dug iteactio. [ ] OR
10 6. A ba has two emegecy souces of owe fo its comutes. Thee is a 95% chace that souce will oeate duig a total owe failue, ad a 80% chace that souce will oeate. Assume the owe souces ae ideedet. What is the obability that at least oe of them will oeate duig a total owe failue? OR OR If P ( A ) 0.3, P ( B ) 0.6. a) Fid P ( A B ) whe A ad B ae ideedet. P ( A B ) P ( A ) P ( B ) ; P ( A B ) P ( A ) + P ( B ) P ( A B ) b) Fid P ( A B ) whe A ad B ae mutually exclusive. P ( A B ) 0; P ( A B ) P A P B 0 B
11 If P ( A ) 0.8, P ( B ) 0.5, ad P ( A B ) 0.9, ae A ad B ideedet evets? Why o why ot? P ( A B ) P ( A ) + P ( B ) P ( A B ) P ( A B ). P ( A B ) P ( A ) P ( B ). P ( A B ) A ad B ae ideedet Die A has oage o oe face ad blue o five faces, Die B has oage o two faces ad blue o fou faces, Die C has oage o thee faces ad blue o thee faces. All ae fai dice. If the thee dice ae olled, fid the obability that exactly two of the thee dice come u oage? P ( O O B ) + P ( O B O ) + P ( B O O )
12 0. A electoic device has fou ideedet comoets. Two of those fou ae ew, ad have a eliability of 0.80 each, oe is old, with 0.75 eliability, ad oe is vey old, ad its eliability is Let Ai { i th comoet is fuctioal }. The P( A ) P( A ) 0.80, P( A3 ) 0.75, P( A4 ) a) Suose that the device wos if all fou comoets ae fuctioal. What is the obability that the device will wo whe eeded? all fou st ad d ad 3d ad 4th itesectio. P( A A A3 A4 ) sice the comoets ae ideedet P( A ) P( A ) P( A3 ) P( A4 ) (0.80) (0.80) (0.75) (0.60) b) Suose that the device wos if at least oe of the fou comoets is fuctioal. What is the obability that the device will wo whe eeded? at least oe eithe st o d o 3d o 4th o 5th uio. P(at least oe) P(oe). oe ot st ad ot d ad ot 3d ad ot 4th ad ot 5th. P( A A A3 A4 ) P( A' A' A3' A4' ) sice the comaies ae ideedet P( A' ) P( A' ) P( A3' ) P( A4' ) (0.0) (0.0) (0.5) (0.40)
13 c) Suose that the fou comoets ae coected as show o the diagam below. Fid the eliability of the system OR
14 . A oal fial exam cotiues util a studet eithe aswes two questios i a ow coectly (ad asses) o aswes two questios i a ow icoectly (ad fails). Suose Alex has obability to aswe ay questio coectly, ideedetly of ay othe questios. What is the obability that Alex would ass the exam? C C C W C C C W C W C C ( C W ) C C W C C W C W C C W C W C W C C ( W C ) W C C +.
15 . Suose Jae has a fai 4-sided die, ad Dic has a fai 6-sided die. Each day, they oll thei dice at the same time (ideedetly) util someoe olls a (as may times as ecessay). (The the eso who did ot oll a does the dishes.) Fid the obability that Jae olls the fist befoe Dic does. ( J D' ) o ( J' D' ) ( J D' ) o ( J' D' ) ( J' D' ) ( J D' ) o OR tu D D' 5 5 J o oe does dishes Dic does dishes J' Jae does dishes game cotiues game eds Jae wis 9 5
16 3. Pove (show) that a) ( Pascal s equatio ).. b)..
17 4. We aleady ow that 0. Pove (show) that m m m 0 m m.
18 The eatig club is hostig a mae-you-ow sudae aty at which the followig ae ovided: Ice Ceam Flavos Chocolate Cooies ceam Stawbey Vailla Toigs Caamel Hot fudge Mashmallow M&M s Nuts Stawbeies a) How may sudaes ae ossible usig oe flavo of ice ceam ad thee diffeet toigs? b) How may sudaes ae ossible usig oe flavo of ice ceam ad fom zeo to six toigs? c) How may diffeet combiatios of flavos of thee scoos of ice ceam ae ossible if it is emissible to mae all thee scoos the same flavo? The umbe of uodeed selectios of 3 objects that ca be made out of 4 objects (eetitios ae allowed) is
Discussion 02 Solutions
STAT 400 Discussio 0 Solutios Spig 08. ~.5 ~.6 At the begiig of a cetai study of a goup of pesos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the fiveyea study, it was detemied
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 informationConsider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample
Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More information= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet
More information2012 GCE A Level H2 Maths Solution Paper Let x,
GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs
More informationUsing Counting Techniques to Determine Probabilities
Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More informationMath 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual
Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, 4.-4.4,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A
More informationICS141: Discrete Mathematics for Computer Science I
Uivesity of Hawaii ICS141: Discete Mathematics fo Compute Sciece I Dept. Ifomatio & Compute Sci., Uivesity of Hawaii Ja Stelovsy based o slides by D. Bae ad D. Still Oigials by D. M. P. Fa ad D. J.L. Goss
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +
More informationMATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES
MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES O completio of this ttoial yo shold be able to do the followig. Eplai aithmetical ad geometic pogessios. Eplai factoial otatio
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationa) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.
Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p
More informationThe number of r element subsets of a set with n r elements
Popositio: is The umbe of elemet subsets of a set with elemets Poof: Each such subset aises whe we pick a fist elemet followed by a secod elemet up to a th elemet The umbe of such choices is P But this
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS
EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - PROGRESSIONS CONTENTS Be able to apply algebaic techiques Aithmetic pogessio (AP): fist
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationOn a Problem of Littlewood
Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationMath 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as
Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1
More informationEXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI
avtei EXAMPLES Solutios of AVTE by AVTE (avtei) lass XI Leade i BSE oachig 1 avtei SHORT ANSWER TYPE 1 Fid the th tem i the epasio of 1 We have T 1 1 1 1 1 1 1 1 1 1 Epad the followig (1 + ) 4 Put 1 y
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationChapter 2 Sampling distribution
[ 05 STAT] Chapte Samplig distibutio. The Paamete ad the Statistic Whe we have collected the data, we have a whole set of umbes o desciptios witte dow o a pape o stoed o a compute file. We ty to summaize
More informationElectron states in a periodic potential. Assume the electrons do not interact with each other. Solve the single electron Schrodinger equation: KJ =
Electo states i a peiodic potetial Assume the electos do ot iteact with each othe Solve the sigle electo Schodige equatio: 2 F h 2 + I U ( ) Ψ( ) EΨ( ). 2m HG KJ = whee U(+R)=U(), R is ay Bavais lattice
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationTaylor Transformations into G 2
Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that
More informationSummary: Binomial Expansion...! r. where
Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly
More informationSupplementary materials. Suzuki reaction: mechanistic multiplicity versus exclusive homogeneous or exclusive heterogeneous catalysis
Geeal Pape ARKIVOC 009 (xi 85-03 Supplemetay mateials Suzui eactio: mechaistic multiplicity vesus exclusive homogeeous o exclusive heteogeeous catalysis Aa A. Kuohtia, Alexade F. Schmidt* Depatmet of Chemisty
More informationGreatest term (numerically) in the expansion of (1 + x) Method 1 Let T
BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value
More informationSOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz
STAT UIU Pctice Poblems # SOLUTIONS Stepov Dlpiz The followig e umbe of pctice poblems tht my be helpful fo completig the homewo, d will liely be vey useful fo studyig fo ems...-.-.- Pove (show) tht. (
More informationAPPLIED THERMODYNAMICS D201. SELF ASSESSMENT SOLUTIONS TUTORIAL 2 SELF ASSESSMENT EXERCISE No. 1
APPIED ERMODYNAMICS D0 SEF ASSESSMEN SOUIONS UORIA SEF ASSESSMEN EXERCISE No. Show how the umeti effiiey of a ideal sigle stage eioatig ai omesso may be eeseted by the equatio ( / Whee is the leaae atio,
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More informationLESSON 15: COMPOUND INTEREST
High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed
More informationn( black) n( joker) n( red joker) Solutions to Unit 1.3 1a) { 7 of diamonds} 1b) { ace of spades, ace of diamonds, ace of hearts, ace of clubs}
Solutios to Uit. a) { 7 of diamods} b) { ace of spades, ace of diamods, ace of hearts, ace of clubs} c) {,,, 5, 6, 7,, 9,0}all of clubs d) {,, 6,,0} of clubs, diamods, hearts, ad spades a) There are 5
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationZero Level Binomial Theorem 04
Zeo Level Biomial Theoem 0 Usig biomial theoem, epad the epasios of the Fid the th tem fom the ed i the epasio of followig : (i ( (ii, 0 Fid the th tem fom the ed i the epasio of (iii ( (iv ( a (v ( (vi,
More informationI PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of
Two Maks Questios I PU MATHEMATIS HAPTER - 08 Biomial Theoem. Epad + usig biomial theoem ad hece fid the coefficiet of y y. Epad usig biomial theoem. Hece fid the costat tem of the epasio.. Simplify +
More informationON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS
Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics
More informationTHE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES
Please cite this atle as: Mhal Matalyck Tacaa Romaiuk The aalysis of some models fo claim pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Sciece 004 Volume 3 Issue pages
More informationInduction: Solutions
Writig Proofs Misha Lavrov Iductio: Solutios Wester PA ARML Practice March 6, 206. Prove that a 2 2 chessboard with ay oe square removed ca always be covered by shaped tiles. Solutio : We iduct o. For
More informationNotes on Hypothesis Testing, Type I and Type II Errors
Joatha Hore PA 818 Fall 6 Notes o Hypothesis Testig, Type I ad Type II Errors Part 1. Hypothesis Testig Suppose that a medical firm develops a ew medicie that it claims will lead to a higher mea cure rate.
More information( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.
Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More informationTopic 4. Representation and Reasoning with Uncertainty
Toic 4 Reresetatio ad Reasoig with Ucertaity Cotets 4.0 Reresetig Ucertaity 4. Probabilistic methods Bayesia PART III 4.2 Certaity Factors CFs 4.3 Demster-Shafer theory 4.4 Fuzzy Logic 4. Probabilistic
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationGRAVITATIONAL FORCE IN HYDROGEN ATOM
Fudametal Joual of Mode Physics Vol. 8, Issue, 015, Pages 141-145 Published olie at http://www.fdit.com/ GRAVITATIONAL FORCE IN HYDROGEN ATOM Uiesitas Pedidika Idoesia Jl DR Setyabudhi No. 9 Badug Idoesia
More informationPutnam Training Exercise Counting, Probability, Pigeonhole Principle (Answers)
Putam Traiig Exercise Coutig, Probability, Pigeohole Pricile (Aswers) November 24th, 2015 1. Fid the umber of iteger o-egative solutios to the followig Diohatie equatio: x 1 + x 2 + x 3 + x 4 + x 5 = 17.
More information4. PERMUTATIONS AND COMBINATIONS Quick Review
4 ERMUTATIONS AND COMBINATIONS Quick Review A aagemet that ca be fomed by takig some o all of a fiite set of thigs (o objects) is called a emutatio A emutatio is said to be a liea emutatio if the objects
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More information9.7 Pascal s Formula and the Binomial Theorem
592 Chapte 9 Coutig ad Pobability Example 971 Values of 97 Pascal s Fomula ad the Biomial Theoem I m vey well acquaited, too, with mattes mathematical, I udestad equatios both the simple ad quadatical
More informationPhysics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving
Physics 11 Chapte 3: Vectos and Motion in Two Dimensions The only thing in life that is achieved without effot is failue. Souce unknown "We ae what we epeatedly do. Excellence, theefoe, is not an act,
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Fall 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to
More informationProfessor Wei Zhu. 1. Sampling from the Normal Population
AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple
More informationDisjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements
Disjoit Sets elemets { x, x, } X =, K Opeatios x Patitioed ito k sets (disjoit sets S, S,, K Fid-Set(x - etu set cotaiig x Uio(x,y - make a ew set by combiig the sets cotaiig x ad y (destoyig them S k
More informationThe Multivariate-t distribution and the Simes Inequality. Abstract. Sarkar (1998) showed that certain positively dependent (MTP 2 ) random variables
The Multivaiate-t distibutio ad the Simes Iequality by Hey W. Block 1, Saat K. Saka 2, Thomas H. Savits 1 ad Jie Wag 3 Uivesity of ittsbugh 1,Temple Uivesity 2,Gad Valley State Uivesity 3 Abstact. Saka
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationChapter 6 Conditional Probability
Lecture Notes o robability Coditioal robability 6. Suppose RE a radom experimet S sample space C subset of S φ (i.e. (C > 0 A ay evet Give that C must occur, the the probability that A happe is the coditioal
More informationCS 70 Second Midterm 7 April NAME (1 pt): SID (1 pt): TA (1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt):
CS 70 Secod Midter 7 April 2011 NAME (1 pt): SID (1 pt): TA (1 pt): Nae of Neighbor to your left (1 pt): Nae of Neighbor to your right (1 pt): Istructios: This is a closed book, closed calculator, closed
More informationPROGRESSION AND SERIES
INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of
More informationSolutions to Problem Set 8
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics fo Compute Science Novembe 21 Pof. Albet R. Meye and Pof. Ronitt Rubinfeld evised Novembe 27, 2005, 858 minutes Solutions to Poblem
More informationELEMENTARY AND COMPOUND EVENTS PROBABILITY
Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 ELEMENTARY AND COMPOUND EVENTS PROBABILITY William W.S. Che Depatmet of Statistics The Geoge Washigto Uivesity Washigto D.C. 003 E-mail: williamwsche@gmail.com
More informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More informationOn randomly generated non-trivially intersecting hypergraphs
O adomly geeated o-tivially itesectig hypegaphs Balázs Patkós Submitted: May 5, 009; Accepted: Feb, 010; Published: Feb 8, 010 Mathematics Subject Classificatio: 05C65, 05D05, 05D40 Abstact We popose two
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationMinimal order perfect functional observers for singular linear systems
Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig
More information( ) = p and P( i = b) = q.
MATH 540 Radom Walks Part 1 A radom walk X is special stochastic process that measures the height (or value) of a particle that radomly moves upward or dowward certai fixed amouts o each uit icremet of
More informationLower Bounds for Cover-Free Families
Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set
More informationECE534, Spring 2018: Final Exam
ECE534, Srig 2018: Fial Exam Problem 1 Let X N (0, 1) ad Y N (0, 1) be ideedet radom variables. variables V = X + Y ad W = X 2Y. Defie the radom (a) Are V, W joitly Gaussia? Justify your aswer. (b) Comute
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 informationCS 330 Discussion - Probability
CS 330 Discussio - Probability March 24 2017 1 Fudametals of Probability 11 Radom Variables ad Evets A radom variable X is oe whose value is o-determiistic For example, suppose we flip a coi ad set X =
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationConditional Probability. Given an event M with non zero probability and the condition P( M ) > 0, Independent Events P A P (AB) B P (B)
Coditioal robability Give a evet M with o zero probability ad the coditio ( M ) > 0, ( / M ) ( M ) ( M ) Idepedet Evets () ( ) ( ) () ( ) ( ) ( ) Examples ) Let items be chose at radom from a lot cotaiig
More informationBINOMIAL THEOREM & ITS SIMPLE APPLICATION
Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB Sllabus : BINOMIAL THEOREM & ITS SIMPLE APPLIATION Biomia Theoem fo a positive itegal ide;
More information11.6 Absolute Convergence and the Ratio and Root Tests
.6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does
More informationKEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow
KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you
More informationIntroduction to the Theory of Inference
CSSM Statistics Leadeship Istitute otes Itoductio to the Theoy of Ifeece Jo Cye, Uivesity of Iowa Jeff Witme, Obeli College Statistics is the systematic study of vaiatio i data: how to display it, measue
More informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationFourier Series and the Wave Equation
Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig
More informationIntermediate Math Circles November 4, 2009 Counting II
Uiversity of Waterloo Faculty of Mathematics Cetre for Educatio i Mathematics ad Computig Itermediate Math Circles November 4, 009 Coutig II Last time, after lookig at the product rule ad sum rule, we
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationPUTNAM TRAINING PROBABILITY
PUTNAM TRAINING PROBABILITY (Last udated: December, 207) Remark. This is a list of exercises o robability. Miguel A. Lerma Exercises. Prove that the umber of subsets of {, 2,..., } with odd cardiality
More informationHomework 5 Solutions
Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the
More informationProbability theory and mathematical statistics:
N.I. Lobachevsky State Uiversity of Nizhi Novgorod Probability theory ad mathematical statistics: Law of Total Probability. Associate Professor A.V. Zorie Law of Total Probability. 1 / 14 Theorem Let H
More informationActivity 3: Length Measurements with the Four-Sided Meter Stick
Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter
More informationThis Technical Note describes how the program calculates the moment capacity of a noncomposite steel beam, including a cover plate, if applicable.
COPUTERS AND STRUCTURES, INC., BERKEEY, CAIORNIA DECEBER 001 COPOSITE BEA DESIGN AISC-RD93 Techical te This Techical te descibes how the ogam calculates the momet caacit of a ocomosite steel beam, icludig
More informationFinite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler
Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio
More informationYou may work in pairs or purely individually for this assignment.
CS 04 Problem Solvig i Computer Sciece OOC Assigmet 6: Recurreces You may work i pairs or purely idividually for this assigmet. Prepare your aswers to the followig questios i a plai ASCII text file or
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationMathematics 116 HWK 21 Solutions 8.2 p580
Mathematics 6 HWK Solutios 8. p580 A abbreviatio: iff is a abbreviatio for if ad oly if. Geometric Series: Several of these problems use what we worked out i class cocerig the geometric series, which I
More informationAn alternating series is a series where the signs alternate. Generally (but not always) there is a factor of the form ( 1) n + 1
Calculus II - Problem Solvig Drill 20: Alteratig Series, Ratio ad Root Tests Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick
More informationMinimization of the quadratic test function
Miimizatio of the quadatic test fuctio A quadatic fom is a scala quadatic fuctio of a vecto with the fom f ( ) A b c with b R A R whee A is assumed to be SPD ad c is a scala costat Note: A symmetic mati
More informationUNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY
UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY Structure 2.1 Itroductio Objectives 2.2 Relative Frequecy Approach ad Statistical Probability 2. Problems Based o Relative Frequecy 2.4 Subjective Approach
More information