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# Mathematics Laboratory MATH M0050

Purdue

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This 30 page Class Notes was uploaded by Elna Vandervort on Saturday September 19, 2015. The Class Notes belongs to MATH M0050 at Purdue University taught by Staff in Fall. Since its upload, it has received 85 views. For similar materials see /class/208035/math-m0050-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

Stat 225 Topic 1 Probability and Sets Introduction to Probability Think Physics not Math A common mistake made by many students who haven7t studied probability or statistics before is to think of it as a math course Probability is an application of math like physics not a particular type of it We will be using math in this course but none of it should be new The hard part of probability is learning when to apply the tools you will learn Of course if you do struggle with the math please come to of ce hours and get help The Purpose of Probability Let7s continue the comparison between probability and physics for another minute Physics uses math to model certain types of behavior in the real world For example if you were to drop a coin from a speci ed height you could calculate the rate at which it dropped using the principlestools you learn in physics ln physics the end result is always the same provided the initial conditions are the same same height same vacuum etc Probability allows us to study situations where the end result varies even when the initial conditions are the same or at least close enough to the same that we cannot distinguish one from another Even if we dont know exactly what the outcome will be probability allows us to make predictions on what the outcome is likely to be For example o If we ip a fair coin 20 times is it likely that it comes up a heads every time o If two parents have a daughter what should they expect the adult height of their child to be ls the answer likely to change if they have a son What if the sex of the child is unknown if for example the baby hasnt been born yet If the stock market has been increasing every day for many days does that mean that it is likely to increase or decrease tomorrow If children are picked at random for a part in the school play how likely is it that two best friends are both picked for major roles By the end of this course you should know how probability can be used to nd answers to all of these questions and many others Course Notation Notation is often one of the trickiest but most important parts of a learning a new subject lmagine trying to make sense of the formula A 7T gtlt r2 the formula for the area of a circle without knowing that 7139 is used to represent irrational number 3141592654 If you do not recognize a certain notation make sure to ask its easy to forget new notation a day or two after its been introduced and it can be hard to understand the topic of the current day when you7re trying to remember what the notation means Review of Sets and Set Notation Set De nition Notation 0 Sets are generally labeled using letters such as 0 An element of a set is represented by letters such as 0 To say that a is an element of the set A we write 0 To de ne a set we need to describe exactly which elements belong to it There are several ways to do this depending on what kinds of elements belong to the set Nonnumeric Sets Non numeric sets list each distinct element within Objects with multiple copies are only listed once For example what is the set of coins made by the US government for circulation When there are a large number of distinct elements it is sometimes possible to use shortcuts to avoid listing every element For example think of a short cut to describe the cards in a standard deck of 52 playing cards Countable Numeric Sets The principles from non numeric sets can always be used on countable numeric sets However countable numeric sets often contain pat terns that can be used to avoid listing every element separately 1 When there are a small xed number of elements in the set we list all ele ments in the set between as in the non numeric case Consequently set of possible results when a 6 sided die is rolled is 2 When there is a countably in nite number of numeric elements displaying a clear pattern we list enough elements at the beginning to establish the pattern and replace the remaining elements with dots The natural counting numbers can be written as Remember that a countably in nite set has in nitely many elements but in theory all of them can be listed 00 When there is a nite number of numeric elements displaying a clear pattern in addition to establishing the pattern it is necessary to show where to stop Thus the last element is listed after the dots Even numbers from 2 to 100 can therefore be written as Uncountable Numeric Sets Remember that uncountable sets are impossible to list For example try to list the numbers from 0 to 1 After 0 no matter how small of a number you choose to come next you can always nd one smaller Since listing the elements of these sets are impossible the idea is to instead focus on where the maximum and minimum for a continuous range are well talk about how to work with multiple continuous ranges momentarily A is used to denote a boundary that includes the boundary point is used to denote a boundary that does not include the boundary point or goes to in nity or negative in nity Examples 1 The range of numbers from 0 to 1 including 1 but not 0 2 All non negative numbers Special Sets There are two special kinds of sets that we need to make note of The empty set Subset Working With Sets Consider two sets A B There are 3 basic operations we can perform on these sets to create new sets 0 To create a new set that consists of all elements in either set we take the 0 To create a new set that consists of the elements in both sets we take the 0 To create a new set that consists of all elements not in set A or B we take the Note Using the compliment requires some knowledge of what the elements not in A are This should be clear from the contest of the problem Laws about Sets Let AB and C be sets Then the following laws are always true Commutative Law Associative Law A BB A A B OA B O AUBBUA AUBUOAUBUO Distributive Law De Morgan s Law A BUOA BUA O AUBCAC BC AUB OAUB AUO A BCACUBC Visualizing Sets Venn Diagrams provide a simple method for Visualizing relationships between sets Union A U B Intersection A B Complement Ac Examples LetA 110B 24207 andO 14710713716719 LetU 120 be the entire set 7 ie Ac U Ac 117 7 20 Determine the elements of the following sets 1 Ac 2 AUB 3 A O 4 O AUBC 53mmm0 aimummm 1A BVUB Draw a Venn Diagram to represent each of the following situations 1 A is a subset of B7 and A is a subset of C 2 AisasubsetofB B Cy Q A C 3 A and B are subsets of Cc Sets and Probability Random Experiment De nition Examples Sets in Probability Sets are used in probability to label groups of possible results of a random experiment Some important sets used in probability include Sample Space Event Outcome Examples Identify the sample space7 the desired event7 and the outcome 1 I ip a coin three times looking for at least 2 heads I actually get a head7 then a tail7 then a second tail 2 I roll a red and a white dice7 looking for the sum to be 10 I actually roll a red six7 and a white four 3 I draw a card from a 52 card deck7 looking for a face card I actually get the ace of diamonds 4 I draw a 5 card poker hand from a 52 card deck7 and l7m looking for a ush all 5 cards are from the same suit Mutually Exclusive Events De nition Note Always keep an eye out for mutually exclusive events they can make many problems much easier Example Are the following events mutually exclusive 1 Getting an even number and getting a number lower than 4 when I roll a 6 sided die 2 If I roll a die two times7 getting an even number on the rst roll7 and the sum of the two dice being 2 3 If I roll a die two times7 getting an even number on the rst roll and getting an odd number on the second roll Frequentist De nition of Probability De ne PE to be the probability that an event E occurs when a given random experiment is performed Suppose that this random experiment is repeated 71 times7 where n is a large number Let be the number of times that the event E occurs Then the frequentist interpretation of probability states that Note PE is a function which takes a set E and translates into a number Axioms of Probability There are three basic assumptions about the function PE needed for probability to make sense These are known as the Axioms of Probability7 or the Nonnegat ivity Certainty Additivity Calculating Probabilities When each element in a sample space is then we can calculate the probability of an event by the probabilitie of each element in the event In other words7 let NE be the number of elements in the set E Then Example Suppose a random experiment consists of rolling a blue and a red dlie7 and recording the sum of both dice 1 How many different numbers can the two dice sum to 2 How many equally likely outcomes are there 3 What is the probability that the sum of the dice is 5 4 What is the probability that the sum of the dice is greater than or equal to 9 5 What is the probability that the sum of the dice is even 6 What is the probability that the sum of the dice is not 67 77 or 8 Properties of Probabilities Probability of the Empty Set Domination Principle Complementation Rule Note The complementatz39on rule is one of the most useful properties in probability Law of Partitions A partition is a group of events7 A1 An in the same sample space where the of all the events is equal to the sample space The Law of Partitions states that the probability of an event B can be found by General Addition Rule When A7 B7 and C are not mutually exclusive7 the probability of their union is Pmum PAUBUO This can be generalized to an arbitrary number of sets using the same principle Examples 1 A survey is sent to a random sample of 50 undergraduates at Purdue with 3 yesno questions Four people responded that they exercise once a week7 are from lndiana7 and have a family member who has also attended Purdue Six people exercise once a week and have a family member at Purdue Seven people are from Indiana and have a family member who has attended Purdue Seventeen people have a family member who has attended Purdue Eight people from Indiana responded that they dont exercise7 and 8 people who exercise are not from Indiana Sixteen people responded no to all three questions a Complete a Venn diagram displaying the number of individuals in each category b If we were to put all the responses into a hat7 what is the probability that i We draw out a response that has a yes for all questions ii Draw out a response with exactly two yeses iii Draw out a response with less than two yeses iv What is the probability we get a response of someone who is from lndiana7 or has a family member who has attended Purdue v What is the probability that the respondent answered yes for at least one ques tion 2 Shade the regions of a 3 way Venn Diagram corresponding to the following regions a AUB b AUB O C A BV C d Cu B m AU 00y 3 If F is the event that a randomly chosen person is female7 and S is the event that the person is single7 then how would you describe the events that the randomly chosen person is a Married b A married female c A single male 4 In a class with 100 students7 50 like math and 60 like statistics 15 like both math and statistics Draw a Venn diagram and answer the following questions a How many students like only math b How many like neither math nor statistics 5 Suppose you roll two fair six sided dlice7 one red and one blue a What is the probability that the two dice will show the same face b What is the probability that the two dice will show different faces C What is the probability that the blue dice shows a 37 and the red die a 4 d What is the probability that one of the dice shows a 37 and the other shows a 4 Review 2 Math 266 Spring 2001 APPLICATIONS You should be able to set up and solve up and solve problems of the types 0 Rate of change of y is proportional to y7 y Icy o Newtonls Law of Cooling7 T kT Te7 Where Te is the temperature of the surrounding environment d 0 Problems in mechanics7 F ma or mdi Sum of external forces Forces on an object that is moving vertically near the surface of the earth d are gravity7 w mg7 and a force proportional to velocity7 so m U tmg lw dt 2 For rocket problems7 the force due to gravity is F d d Use iv vi and nd the velocity as a function of the height above the surface of the gecarth o Mixing problems Let Qt be the amount of Whatever is in solution and let Vt be the amount of solution CM 15 t V t mysocx co 0 Rate of Q coming in is Concentration in Rate of solution in Rate of Q going out is Concentration at time 23 Rate of solution out dQ E Caution if Rate of solution in 7E Rate of solution out then you must The concentration of Q at time t is then Ct Rate of Q coming in Rate of Q going out d solve for Vt from E Rate of solution in Rate of solution out If the in rates and out rates are constant7 then Vt V0 kt Where k is the difference of the rates 2 MA 266 SPR 01 REVIEW 2 PRACTICE QUESTIONS 1 Suppose y is proportional to y y0 47 and y2 2 Find y in terms of t For what value oft does yt 3 2 A thermometer reads 360 when it is moved into a 700 room Five minutes later the thermometer reads 500 Find the thermometer reading t minutes after it is moved into the room What will it read ten minutes after it is moved into the room 3 Determine the vertical velocity of a 1284b parachutist t seconds after jumping from an airplane that is ying slowly and horizontally at an altitude of 5000 feet Assume that air resistance is eight times the speed and ignore horizontal motion 4 A rocket is launched from earth with initial vertical velocity Find a formula that relates the velocity of the rocket and its height What is the maximum height of the rocket Assume that gravity is the only force acting on the rocket and that the mass of the rocket is constant 5 A 500gal tank contains 200 gal of brine with salt concentration of 2 oz gal Pure water ows into the tank at a rate of 10 gal min7 while the mixture ows out of the tank at a rate of 5 gal min Find the salt concentration in the tank at the time the tank becomes completely lled 6 Consider the differential equation yy 27 t 2 07 Hint draw a rough graph of F yy 2 to analyse where y is positive and negative a Sketch the graph of the solution of the differential equation for t 2 0 with each of the initial values y0 237 y0 07 y0 237 y 43 y0 2 y0 83 b What are the equilibrium solutions c Which equilibrium solutions are stable d For which intervals of y is the graph of yt increasing e For which intervals of y is the graph of yt concave upward 7 Solve the initial value problem 3 2y 07 y0 3 8 Solve the initial value problem 3 2y 17 y0 3 9 Solve the initial value problem 3 y0 1 What is the largest open interval on which the solution is valid 3 10 Solve the initial value problem 3 yy 17 y0 2 What is the largest open interval on Which the solution is valid 11 Solve the initial value problem y ylac7 y1 47 y 1 2 12 Solve the initial value problem y y 2y7 y0 e y 0 1 13 For each of the initial value problems determine the largest interval for Which a unique solution is guaranteed a y 21 211 0 b y tan ty sec 75 y0 0 C y my y0 1 ltdgtltt4gty39 ty 21 14 For each of the initial value problems determine all initial points to7 yo for Which a unique solution is guaranteed in some interval to h lt t lt to h a y 762 92 W0 yo b y 762 312 31050 yo C 2 7621 We yo 01 y 7613212 We yo 4 MA 266 SPR 01 REVIEW 2 PRACTICE QUESTION ANSWERS 1 y 4elt1n05gtt2 t 21128 m 083 2 T 70 34elt1nlt1017gtgtt5 T10 m 5820 3 1116 16 1637 4 v2 Rg wmax B when U 0 5 C60 032 OZgal 6 y gt01feitherylt00rygt2 y lt01f0ltylt2 a For y0 23 yt increases asympt to line y 0 For y0 0 stays on y 0 line For y0 23 or 43 yt decreases asympt to y 0 line For y0 2 stays on y 2 line For y0 83 decreases asympt to liney2 by0 y2 Cy0 dylt0 ygt2 e0ltylt1 ygt2 7y3e 2t 8y12e2t 1 9 t 1 y t1 gt 10 2 tlt12 n y 2et7 11y3102 12 y 6 605 7T 7 14 a All t0y0 b All t0y0 except 00 c All t0y0 with yo 74 0 d All to yo with yo 74 0 13 atgt0 b glttlt c 3lttlt2d 4lttlt0 MA 266 Spring 2001 REVIEW 6 LAPLACE TRANSFORMS You should be able to express functions that are de ned by different formulas 07 t lt c on different intervals in terms of the step functions uct 1 t gt c After the vector t has been de ned7 the MATLAB statement 6 gt 0 gives value zero for t g c and value one for t gt c The statement 6 gt c can then be used for the step function uct to graph functions that involve step functions EXAMPLE To obtain the graph of y uwt sint 7T U37rt sint 377 gvaph u grown 7 pl 7 Mammmap The jump caused by a step function Will appear as a nearly vertical line on your plot EXAMPLE To obtain the graph of y 1 u1t 2u2t U3t Gvaph uly1 Hun 2U72 U73 2 You should be able to use the de nition of the Laplace transform as an im proper integral to evaluate Laplace transforms You should be able to use Table 621 in the text to evaluate Laplace transforms and inverse Laplace transforms 0 Formula 13 tells us that L uctft 0 e CSFs7 where Fs L To use Formula 13 to evaluate L uctgt we rst note that the factor uct corresponds to the factor 6 05 in the Laplace transform We then note that we need to evaluate the Laplace transform of f 7 where ft c gt We can substitute t c for t in the formula ft c gt to obtain the formula for ft and then evaluate Fs L EXAMPLE Evaluate L u1tt2 SOLUTION We rst note that Formula 13 applies with c 1 Also7 the function f in Formula 13 must satisfy ft 1 7 Substituting t 1 for t in this formula7 we obtain ft t 12 t2 216 1 Table 621 then gives 2 2 1 2 is 2 2 1 Lft8 3S 2ESO U1LLLL 0 Formula 13 also says L 1e CSFs uctft c where ft L 1 To use Formula 13 to evaluate L 1 e CSFs7 we rst note that the factor 6 05 corresponds to the factor uct in the inverse Laplace transform We then use Table 621 to evalaute ft L 1 We then substitute t c for t in the formula for ft to obtain ft 0 EXAMPLE Evaluate L 1 6 25 8 12 SOLUTION We rst note that Formula 13 applies with c 2 Table 621 1 gives ft L 1 tet Substitution of t 2 for t in the formula 8 1 for ft gives ft 2 t 2et 27 so L 1e25812 u2tt 2et 2 PARTIAL FRACTIONS The partial fraction expansion of a rational function P Q consists of the sum of a polynomial and terms that have a power of a factor of Q in the denominators To obtain the partial fraction expansion of P Q 0 Check the degrees of the numerator and denominator If deg P 2 deg Q diuide P by Q to obtain polynomials S and R with degR lt degQ and Rim 6293 If degP lt deg Q then 5510 0RU P U and it is not necessary to divide 0 Factor the denominator into powers of distinct linear terms and powers of irreducible quadratic terms This factorization determines the form of the empansion of R Q SU A theorem of algebra guarantees that any polynomial with real number coef cients can be factored into powers of linear terms and powers of irreducible quadratic terms The linear factors can be written in the form on r7 where r is a zero of the denominator The quadratic 132 bat c is irreducible if it has no real zeros That is7 if b2 4ac lt 0 o If contains emactly m identical linear factors on r the expansion of RQ contains a sum of the form A1 A2 Am Jr Jr 7 w r av r2 at rm where A1 A2 Am are unknown constants o If contains emactly n identical irreducible quadratic factors 102 bac c the expansion of R Q contains a sum of the form B1 UCl B2 UCg 1502 bat c 1562 bat c2 1562 bat cn7 where B1 C1Bg Cg Bn7 Cm are unknown constants Each distinct factor of Q contributes a term or terms of the type indicated to the expansion of R Q In each case7 the denominators are powers of the factors7 where the powers run from the rst power up to the power of the factor in Q Terms that have powers of linear factors in the denominator have numerators that are numbers Terms that have powers of irreducible quadratic factors in the denominator have numerators of the form numberc number 0 Complete the partial fraction expansion by solving for the values of the un known constants A theorem of algebra guarantees a unique solution for the values MA 266 Spring 2001 REVIEW 6 PRACTICE QUESTIONS Oxpr XJKI 10 11 12 13 14 15 16 17 18 19 evaluate the Laplace transform of ft ft zW 3y2yomm4y yW y5 wm1w m t L 0 u1t Tet77 COS27 d7 L2 L sin2t cos2t L e t1 cos3t L e2tt3 sint l1qtUQ t2 a 0 tltL 2 1gtltz 0 tgz f l 17 y 2 y 0 17 y 0 17 Ly t Use the de nition of the Laplace transform as an improper integral to a 0gtltz e t t2 2 6 MA 266 Spring 2001 REVIEW 6 PRACTICE QUESTION ANSWERS 2 2 8 10 11 12 13 14 15 16 17 18 19 b 7s1t kgbe 3 83 8 824824 1 81 m 6 1 m s 6725 i 8 83 82 1 2675 e725 2 4 8 gt 8 8 4 m 53 282s cos 2t s1n2t 2tcos2t sin2t t2et T W a 2 6 22gt U1 b 1 C0st 1 675 1 6t 2 t 1L 7 217 2 T d7 0 2 2 L wk y m m f 675 1 1 672s1 6725111 81 51 MA 266 Fall 00 REVIEW 6 LAPLACE TRANSFORMS You should be able to express functions that are de ned by different formulas 07 t lt c on different intervals in terms of the step functions uct 1 t gt c After the vector t has been de ned7 the MATLAB statement 6 gt 0 gives value zero for t g c and value one for t gt c The statement 6 gt c can then be used for the step function uct to graph functions that involve step functions EXAMPLE To obtain the graph of y uwt sint 7T U37rt sint 377 gvaph u Himsmu 7 pl 7 u73pismk3pl The jump caused by a step function Will appear as a nearly vertical line on your plot EXAMPLE To obtain the graph of y 1 u1t 2u2t U3t Gvaph uly1 M710 2U72 U73 2 You should be able to use the de nition of the Laplace transform as an im proper integral to evaluate Laplace transforms You should be able to use Table 621 in the text to evaluate Laplace transforms and inverse Laplace transforms 0 Formula 13 tells us that L uctft 0 e CSFs7 where Fs Lft To use Formula 13 to evaluate L uctgt we rst note that the factor uct corresponds to the factor 6 05 in the Laplace transform We then note that we need to evaluate the Laplace transform of f 7 where ft c gt We can substitute t c for t in the formula ft c gt to obtain the formula for ft and then evaluate Fs L EXAMPLE Evaluate L u1tt2 SOLUTION We rst note that Formula 13 applies with c 1 Also7 the function f in Formula 13 must satisfy ft 1 7 Substituting t 1 for t in this formula7 we obtain ft t 12 t2 216 1 Table 621 then gives 2 2 1 2 is 2 2 1 Lft8 3S 2ESO U1LLLL 0 Formula 13 also says L 1e CSFs uctft c where ft L 1 To use Formula 13 to evaluate L 1 e CSFs7 we rst note that the factor 6 05 corresponds to the factor uct in the inverse Laplace transform We then use Table 621 to evalaute ft L 1 We then substitute t c for t in the formula for ft to obtain ft 0 EXAMPLE Evaluate L 1 6 25 8 12 SOLUTION We rst note that Formula 13 applies with c 2 Table 621 1 gives ft L 1 tet Substitution of t 2 for t in the formula 8 1 for ft gives ft 2 t 2et 27 so L 1e25812 u2tt 2et 2 PARTIAL FRACTIONS The partial fraction expansion of a rational function P Q consists of the sum of a polynomial and terms that have a power of a factor of Q in the denominators To obtain the partial fraction expansion of P Q 0 Check the degrees of the numerator and denominator If deg P 2 deg Q divide P by Q to obtain polynomials S and R with degR lt deg Q and 1393 R193 C25 QW If degP lt deg Q then Sx 0Rx Px and it is not necessary to divide 0 Factor the denominator into powers of distinct linear terms and powers of irreducible quadratic terms This factorization determines the form of the expansion of R Q A theorem of algebra guarantees that any polynomial with real number coef cients can be factored into powers of linear terms and powers of irreducible quadratic terms The linear factors can be written in the form x r7 where r is a zero of the denominator The quadratic ax2 bx c is irreducible if it has no real zeros That is7 if b2 4ac lt 0 o If contains exactly m identical linear factors x r the expansion of RQ contains a sum of the form Sx A1 A2 Am Jr Jr 7 x rm 013 rzc r2 where A1 A2 Am are unknown constants o If contains exactly n identical irreducible quadratic factors ax2 bx c the expansion of R Q contains a sum of the form B1 UCl B2 UCg ax2 bx c ax2 bx c2 ax2 bx c 7 where B1 C1Bg Cg Bn7 Cm are unknown constants Each distinct factor of Q contributes a term or terms of the type indicated to the expansion of R Q In each case7 the denominators are powers of the factors7 where the powers run from the rst power up to the power of the factor in Q Terms that have powers of linear factors in the denominator have numerators that are numbers Terms that have powers of irreducible quadratic factors in the denominator have numerators of the form numberx number 0 Complete the partial fraction expansion by solving for the values of the un known constants A theorem of algebra guarantees a unique solution for the values 4 MA 266 SPR 00 REVIEW 6 PRACTICE QUESTIONS 1 m kwN XJKI 10 11 12 13 14 15 16 17 18 19 evaluate the Laplace transform of ft ft y 3y2y0mmL lt y L 0 u1t Tet T COS27 d7 L2 L sin2t C0s2t L e t1 C0s3t L e2tt3 sint l1qtUQ t2 a 0 tltL 2 1gtltz 0 tgz l l 17 y y y57MWLyWDZMWDLywm L y t Use the de nition of the Laplace transform as an improper integral to a 0gtltz e t t2 2 MA 266 SPR 00 REVIEW 6 PRACTICE QUESTION ANSWERS 2 2 8 10 11 12 13 14 15 16 17 18 19 b 7s1t 51120er dt 191310 83 8 824 824 1 81 s1s129 6 s 22 1 s 6725 i s 83 82 1 26 5 e725 2 4 8 gt 8 8 4 82 382 83 2828 cos 275 sin2t e2tcos2t sin2t t2et T 1 u2t ltt 2 t 2V u1t1 C0st 1 675 1 6t 2 t 1 2 1 2 739 0 t 7 ET 6 d7 ft f5 fte 5tdt ff e te stdt 67s1b 672s1 s 1 672s1 81

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