Mathematical Notation Math Finite Mathematics

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1 Mathematical Notatio Math 60 - Fiite Mathematics Use Word or WordPerfect to recreate the followig documets. Each article is worth 0 poits ad should be ed to the istructor at james@richlad.edu. If you use Microsoft Works to create the documets, the you must prit it out ad gie it to the istructor as he ca t ope those files. Type your ame at the top of each documet. Iclude the title as part of what you type. The lies aroud the title are't that importat, but if you will type at the begiig of a lie ad hit eter, both Word ad WordPerfect will draw a lie across the page for you. For expressios or equatios, you should use the equatio editor i Word or WordPerfect. The istructor used WordPerfect ad a 4 pt Times New Roma fot with stadard 0.75" margis, so they may ot look exactly the same as your documet. The equatios were created usig 6 pt fot, but feel free to use a smaller fot. For idiidual symbols (µ, σ, etc) withi the text of a setece, you ca isert symbols. I Word, use "Isert / Symbol" ad choose the Symbol fot. For WordPerfect, use Ctrl-W ad choose the Greek set. Howeer, it's ofte easier to just use the equatio editor as expressios are usually more complex tha just a sigle symbol. If there is a equatio, put both sides of the equatio ito the same equatio editor box istead of creatig two objects. There are istructios o how to use the equatio editor i a separate documet or o the website. Be sure to read through the help it proides. There are some examples at the ed that walk studets through the more difficult problems. You will wat to read the hadout o usig the equatio editor if you hae ot used this software before. If you fail to type your ame o the page, you will lose poit. Do't type the hits or the remiders at the bottom of each page. These otatios are due at the begiig of class o the day of the exam for that chapter. That is, the chapter 3 otatio is due o the day of the chapter 3 test. Late work will be accepted but will lose 20% of its alue per class period.

2 Chapter 3 - Fiace Simple Iterest I PRT I Iterest P Pricipal R Rate T Time Compoud Iterest ( ) A P + i A Amout P Pricipal i Periodic rate Number of periods Future Value Auities FV ( i) + PMT i FV Future alue PMT Paymet i Periodic rate Number of periods Preset Value Auities PMT i PV ( + i) PV Preset alue PMT Paymet i Periodic rate Number of periods

3 Chapter 4 - Systems of Equatios, Matrices Matrix Additio Determiat of a matrix Augmeted matrix i reduced row-echelo form Solig a system of liear equatios usig matrix ierses AX B X A B Leotief Iput-Output Model ( ) X I M D X Output matrix M Techology Matrix D Demad Matrix

4 Chapter 5 - Liear Programmig Fudametal Theorem of Liear Programmig If there is a solutio to a liear programmig problem, the it will occur at oe or more corer poits of the feasible regio or o the boudary betwee two corer poits. Hit, place followig two systems ito a matrix without brackets. Defie the matrix row spacig to be 00% ad the matrix colum spacig to be 50%. System of liear iequalities 3x + 4y 36 3x + 2y 30 xy 00 Iitial system for a stadard maximizatio problem x + x2 + x3 + s 00 40x + 20x2 + 30x3 + s x + 2x2 + x3 + s x 300x 200x + P Be sure to reset the matrix row spacig to 50% ad the matrix colum spacig to 00%. Iitial tableau for a stadard maximizatio problem

5 Chapter 6 - Sets ad Coutig If A {, 2, 4, 6 } ad B { 2, 3, 5 }, the the uio of the sets is A B {, 2,3, 4,5,6}... the itersectio of the sets is A B { 2} Fudametal Coutig Priciple The total umber of ways that two eets ca happe is foud by multiplyig together the umber of ways that each eet ca happe. Permutatios A permutatio is a arragemet of objects without repetitio but with regard to order. P r! ( r)! Combiatios A combiatio is a arragemet of objects without repetitio ad without regard to order. C r! r r! ( r)!

6 Chapter 7 - Probability Probability formulas Additio Rule ( ) ( ) + ( ) ( ) P A B P A P B P A B Multiplicatio Rule ( ) ( ) ( ) P A B P A P B A Coditioal Probability ( ) P A B ( B) P( B) P A Complemet of a Eet ( ) P( E) P E Expected alue ( ) x p( x ) E x k k k Decisio Theory Expected alue (Bayesia) criterio. Fid the expected alue uder each actio ad choose the actio with the largest expected alue. Maximax criterio. Fid the maximum payoff uder each actio ad the choose the actio with the largest best case sceario. Maximi criterio. Fid the miimum payoff uder each actio ad the choose the actio with the largest worst case sceario. Miimax criterio. Fid the opportuistic loss for each state of ature. The fid the maximum opportuistic loss for each actio ad choose the actio with the smallest maximum loss.

7 Chapter 8 - Statistics Biomial Probabilities A biomial experimet is a fixed umber of idepedet trials each haig exactly two possible outcomes. P x p q p q x x x ( ) ; + total umber of trials p probability of success o a sigle trial q probability of failure o a sigle trial x umber of successes out of trials. Z-scores A stadardized score is foud by takig the alue, subtractig the mea, ad diidig by the stadard deiatio. z x µ σ Normal Distributios A ormal distributio refers to a symmetric, bell-shaped cure. Approximately 68% of the alues lie withi oe stadard deiatio of the mea, 95% of the alues lie withi two stadard deiatios of the mea, ad 99.7% of the alues lie withi three stadard deiatios of the mea. The area uder the etire cure is.

8 Chapter 9 - Game Theory Assume that the game matrix is a b M c d The solutio to a two player, zero-sum, o-strictly determied game is P Q d c a b D D d b D a c D D a+ d b+ c where ( ) ( ) Liear Programmig Problem Assume that M has all positie etries. The optimal row player solutio is foud by solig this liear programmig problem z x+ x2 x ax + cx2 bx + dx2 x, x 0 Miimize: where ad Subject to: 2 The optimal colum player solutio is foud by solig this liear programmig problem. z y+ y2 y ay + by2 cy + dy2 y, y 0 Maximize: where ad Subject to: 2 p q x y 2 2 p 2 q 2

9 Chapter 0 - Marko Chais S is a state matrix, P is the trasitio matrix Regular Marko chais State Matrices S S P 0 S S P S P 2 0 S S P S P Steady State Matrix S SP If the Marko chai is regular, the there is a limitig matrix is the Steady State matrix. P P where each row Absorbig Marko Chais I 0 P R Q Fudametal Matrix F The elemet i row R, colum C of the fudametal matrix represets the expected umber of times you will sped i state trasiet C of the system if you start i trasiet state R before eterig a absorbig state. ( ) F I Q Limitig Matrix I 0 P P FR 0 FR The elemet i row R, colum C of the matrix represets the log term probability of edig up i absorbig state C if you started i trasiet state R.

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