# Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Size: px
Start display at page:

Transcription

1 Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive distriution function (cdf). 3. Be le to eplin why we use proility density for continuous rndom vriles. 2 Introduction We now turn to continuous rndom vriles. All rndom vriles ssign numer to ech outcome in smple spce. Wheres discrete rndom vriles tke on discrete set of possile vlues, continuous rndom vriles hve continuous set of vlues. Computtionlly, to go from discrete to continuous we simply replce sums y integrls. It will help you to keep in mind tht (informlly) n integrl is just continuous sum. Emple. Since time is continuous, the mount of time Jon is erly (or lte) for clss is continuous rndom vrile. Let s go over this emple in some detil. Suppose you mesure how erly Jon rrives to clss ech dy (in units of minutes). Tht is, the outcome of one tril in our eperiment is time in minutes. We ll ssume there re rndom fluctutions in the ect time he shows up. Since in principle Jon could rrive, sy, 3.43 minutes erly, or 2.7 minutes lte (corresponding to the outcome -2.7), or t ny other time, the smple spce consists of ll rel numers. So the rndom vrile which gives the outcome itself hs continuous rnge of possile vlues. It is too cumersome to keep writing the rndom vrile, so in future emples we might write: Let T = time in minutes tht Jon is erly for clss on ny given dy. 3 Clculus Wrmup While we will ssume you cn compute the most fmilir forms of derivtives nd integrls y hnd, we do not epect you to e clculus whizzes. For tricky epressions, we ll let the computer do most of the clculting. Conceptully, you should e comfortle with two views of definite integrl.. f() d = re under the curve y = f(). 2. f() d = sum of f() d.

2 8.05 clss 5, Continuous Rndom Vriles, Spring The connection etween the two is: n re sum of rectngle res = f( )Δ + f( 2 )Δ f( n )Δ = f( i )Δ. As the width Δ of the intervls gets smller the pproimtion ecomes etter. y y = f() y Are = f( i ) y = f() 0 2 n Are is pproimtely the sum of rectngles Note: In clculus you lerned to compute integrls y finding ntiderivtives. This is importnt for clcultions, ut don t confuse this method for the reson we use integrls. Our interest in integrls comes primrily from its interprettion s sum nd to much lesser etent its interprettion s re. 4 Continuous Rndom Vriles nd Proility Density Functions A continuous rndom vrile tkes rnge of vlues, which my e finite or infinite in etent. Here re few emples of rnges: [0, ], [0, ), (, ), [, ]. Definition: A rndom vrile X is continuous if there is function f() such tht for ny c d we hve d P (c X d) = f() d. () c The function f() is clled the proility density function (pdf). The pdf lwys stisfies the following properties:. f() 0 (f is nonnegtive). 2. f() d = (This is equivlent to: P ( < X < ) = ). The proility density function f() of continuous rndom vrile is the nlogue of the proility mss function p() of discrete rndom vrile. Here re two importnt differences:. Unlike p(), the pdf f() is not proility. You hve to integrte it to get proility. (See section 4.2 elow.) 2. Since f() is not proility, there is no restriction tht f() e less thn or equl to.

3 8.05 clss 5, Continuous Rndom Vriles, Spring Note: In Property 2, we integrted over (, ) since we did not know the rnge of vlues tken y X. Formlly, this mkes sense ecuse we just define f() to e 0 outside of the rnge of X. In prctice, we would integrte etween ounds given y the rnge of X. 4. Grphicl View of Proility If you grph the proility density function of continuous rndom vrile X then P (c X d) = re under the grph etween c nd d. f() P (c X d) Think: Wht is the totl re under the pdf f()? c d 4.2 The terms proility mss nd proility density Why do we use the terms mss nd density to descrie the pmf nd pdf? Wht is the difference etween the two? The simple nswer is tht these terms re completely nlogous to the mss nd density you sw in physics nd clculus. We ll review this first for the proility mss function nd then discuss the proility density function. Mss s sum: If msses m, m 2, m 3, nd m 4 re set in row t positions, 2, 3, nd 4, then the totl mss is m + m 2 + m 3 + m 4. m m 2 m 3 m We cn define mss function p() with p( j ) = m j for j =, 2, 3, 4, nd p() = 0 otherwise. In this nottion the totl mss is p( ) + p( 2 ) + p( 3 ) + p( 4 ). The proility mss function ehves in ectly the sme wy, ecept it hs the dimension of proility insted of mss. Mss s n integrl of density: Suppose you hve rod of length L meters with vrying density f() kg/m. (Note the units re mss/length.) Δ i n = L mss of i th piece f( i )Δ

4 8.05 clss 5, Continuous Rndom Vriles, Spring If the density vries continuously, we must find the totl mss of the rod y integrtion: L totl mss = f() d. This formul comes from dividing the rod into smll pieces nd summing up the mss of ech piece. Tht is: n totl mss f( i ) Δ i= In the limit s Δ goes to zero the sum ecomes the integrl. The proility density function ehves ectly the sme wy, ecept it hs units of proility/(unit ) insted of kg/m. Indeed, eqution () is ectly nlogous to the ove integrl for totl mss. While we re on physics kick, note tht for oth discrete nd continuous rndom vriles, the epected vlue is simply the center of mss or lnce point. Emple 2. Suppose X hs pdf f() = 3 on [0, /3] (this mens f() = 0 outside of [0, /3]). Grph the pdf nd compute P (. X.2) nd P (. X ). nswer: P (. X.2) is shown elow t left. We cn compute the integrl:.2.2 P (. X.2) = f() d = 3 d =.3... Or we cn find the re geometriclly: re of rectngle = 3. =.3. 0 P (. X ) is shown elow t right. Since there is only re under f() up to /3, we hve P (. X ) = 3 (/3.) =.7. 3 f() 3 f()..2 /3. /3 P (. X.2) P (. X ) Think: In the previous emple f() tkes vlues greter thn. Why does this not violte the rule tht proilities re lwys etween 0 nd? Note on nottion. We cn define rndom vrile y giving its rnge nd proility density function. For emple we might sy, let X e rndom vrile with rnge [0,]

5 8.05 clss 5, Continuous Rndom Vriles, Spring nd pdf f() = /2. Implicitly, this mens tht X hs no proility density outside of the given rnge. If we wnted to e solutely rigorous, we would sy eplicitly tht f() = 0 outside of [0,], ut in prctice this won t e necessry. Emple 3. Let X e rndom vrile with rnge [0,] nd pdf f() = C 2. Wht is the vlue of C? nswer: Since the totl proility must e, we hve f() d = C 2 d =. 0 0 By evluting the integrl, the eqution t right ecomes C/3 = C = 3. Note: We sy the constnt C ove is needed to normlize the density so tht the totl proility is. Emple 4. Let X e the rndom vrile in the Emple 3. Find P (X /2). /2 /2 nswer: P (X /2) = 3 2 d = 3 = Think: For this X (or ny continuous rndom vrile): Wht is P ( X )? Wht is P (X = 0)? Does P (X = ) = 0 men tht X cn never equl? In words the ove questions get t the fct tht the proility tht rndom person s height is ectly 5 9 (to infinite precision, i.e. no rounding!) is 0. Yet it is still possile tht someone s height is ectly 5 9. So the nswers to the thinking questions re 0, 0, nd No. 4.3 Cumultive Distriution Function The cumultive distriution function (cdf) of continuous rndom vrile X is defined in ectly the sme wy s the cdf of discrete rndom vrile. F () = P (X ). Note well tht the definition is out proility. When using the cdf you should first think of it s proility. Then when you go to clculte it you cn use F () = P (X ) = f() d, where f() is the pdf of X. Notes:. For discrete rndom vriles, we defined the cumultive distriution function ut did

6 8.05 clss 5, Continuous Rndom Vriles, Spring not hve much occsion to use it. The cdf plys fr more prominent role for continuous rndom vriles. 2. As efore, we strted the integrl t ecuse we did not know the precise rnge of X. Formlly, this still mkes sense since f() = 0 outside the rnge of X. In prctice, we ll know the rnge nd strt the integrl t the strt of the rnge. 3. In prctice we often sy X hs distriution F () rther thn X hs cumultive distriution function F (). Emple 5. Find the cumultive distriution function for the density in Emple 2. nswer: For in [0,/3] we hve F () = f() d = 3 d = Since f() is 0 outside of [0,/3] we know F () = P (X ) = 0 for < 0 nd F () = for > /3. Putting this ll together we hve 0 if < 0 F () = 3 if 0 /3 if /3 <. Here re the grphs of f() nd F (). 3 f() F () /3 /3 Note the different scles on the verticl es. Rememer tht the verticl is for the pdf represents proility density nd tht of the cdf represents proility. Emple 6. Find the cdf for the pdf in Emple 3, f() = 3 2 on [0, ]. Suppose X is rndom vrile with this distriution. Find P (X < /2). nswer: f() = 3 2 on [0,] F () = 3 2 d = 3 on [0,]. Therefore, 0 0 if < 0 F () = 3 if 0 if < Thus, P (X < /2) = F (/2) = /8. Here re the grphs of f() nd F (): 3 f() F ()

7 8.05 clss 5, Continuous Rndom Vriles, Spring Properties of cumultive distriution functions Here is summrry of the most importnt properties of cumultive distriution functions (cdf). (Definition) F () =P (X ) 2. 0 F () 3. F () is non-decresing, i.e. if then F () F (). 4. lim F () = nd lim F () =0 5. P ( X ) =F () F () 6. F () =f(). Properties 2, 3, 4 re identicl to those for discrete distriutions. The grphs in the previous emples illustrte them. Property 5 cn e seen lgericlly: f() d = f() d + f() d f() d = f() d f() d P ( X ) =F () F (). Property 5 cn lso e seen geometriclly. The ornge region elow represents F () nd the striped region represents F (). Their difference is P ( X ). P ( X ) Property 6 is the fundmentl theorem of clculus. 4.5 Proility density s drtord We find it helpful to think of smpling vlues from continuous rndom vrile s throwing drts t funny drtord. Consider the region underneth the grph of pdf s drtord. Divide the ord into smll equl size squres nd suppose tht when you throw drt you re eqully likely to lnd in ny of the squres. The proility the drt lnds in given region is the frction of the totl re under the curve tken up y the region. Since the totl re equls, this frction is just the re of the region. If X represents the -coordinte of the drt, then the proility tht the drt lnds with -coordinte etween nd is just P ( X ) = re under f() etween nd = f() d.

8 MIT OpenCourseWre Introduction to Proility nd Sttistics Spring 204 For informtion out citing these mterils or our Terms of Use, visit:

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

### Chapter 9 Definite Integrals

Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

### Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk

Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions

### Chapter 6 Continuous Random Variables and Distributions

Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### 2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

### Interpreting Integrals and the Fundamental Theorem

Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

### The practical version

Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

### Fundamental Theorem of Calculus

Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under

### Experiments, Outcomes, Events and Random Variables: A Revisit

Eperiments, Outcomes, Events nd Rndom Vriles: A Revisit Berlin Chen Deprtment o Computer Science & Inormtion Engineering Ntionl Tiwn Norml University Reerence: - D. P. Bertseks, J. N. Tsitsiklis, Introduction

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

### Chapter 8.2: The Integral

Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

### Section 4: Integration ECO4112F 2011

Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### Review of Gaussian Quadrature method

Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### Math 131. Numerical Integration Larson Section 4.6

Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

### 0.1 THE REAL NUMBER LINE AND ORDER

6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.

### and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

### 2.4 Linear Inequalities and Interval Notation

.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

### Section 6: Area, Volume, and Average Value

Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### The Trapezoidal Rule

_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### Topics Covered AP Calculus AB

Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

### Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17

EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,

### Time in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.

4. Are under Curve A cr is trveling so tht its speed is never decresing during 1-second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37

### Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 17

CS 70 Discrete Mthemtics nd Proility Theory Summer 2014 Jmes Cook Note 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion, y tking

### Chapter 1: Logarithmic functions and indices

Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

### 4.1. Probability Density Functions

STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

### 5.7 Improper Integrals

458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

### n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

### 5.1 How do we Measure Distance Traveled given Velocity? Student Notes

. How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis

### Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

### Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that

Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under

### AB Calculus Review Sheet

AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

### ( ) as a fraction. Determine location of the highest

AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

### Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

### Appendix 3, Rises and runs, slopes and sums: tools from calculus

Appendi 3, Rises nd runs, slopes nd sums: tools from clculus Sometimes we will wnt to eplore how quntity chnges s condition is vried. Clculus ws invented to do just this. We certinly do not need the full

### Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

### Chapter 6 Techniques of Integration

MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

### ( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

### 1 Probability Density Functions

Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our

### An Overview of Integration

An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is

### 7. Indefinite Integrals

7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find

### Linear Inequalities. Work Sheet 1

Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges \$ 15 per week plus \$ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### Chapter 0. What is the Lebesgue integral about?

Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

### What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

### Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

### Math 259 Winter Solutions to Homework #9

Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

### Integration. 148 Chapter 7 Integration

48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

### ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### Chapter 2. Random Variables and Probability Distributions

Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

### 4.6 Numerical Integration

.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

### Mathematics Number: Logarithms

plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

### 5: The Definite Integral

5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

### The Evaluation Theorem

These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for in-clss presenttion nd should not

### Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

### Lecture 2: January 27

CS 684: Algorithmic Gme Theory Spring 217 Lecturer: Év Trdos Lecture 2: Jnury 27 Scrie: Alert Julius Liu 2.1 Logistics Scrie notes must e sumitted within 24 hours of the corresponding lecture for full

### y = f(x) This means that there must be a point, c, where the Figure 1

Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile

### Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### 7.2 The Definite Integral

7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

### Designing Information Devices and Systems I Spring 2018 Homework 7

EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should

### Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

### Calculus AB. For a function f(x), the derivative would be f '(

lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:

### 13.4 Work done by Constant Forces

13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

### 5.4. The Fundamental Theorem of Calculus. 356 Chapter 5: Integration. Mean Value Theorem for Definite Integrals

56 Chter 5: Integrtion 5.4 The Fundmentl Theorem of Clculus HISTORICA BIOGRAPHY Sir Isc Newton (64 77) In this section we resent the Fundmentl Theorem of Clculus, which is the centrl theorem of integrl

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### 10 Vector Integral Calculus

Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

### The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F

### Lecture 3: Equivalence Relations

Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts

### Review of Probability Distributions. CS1538: Introduction to Simulations

Review of Proility Distriutions CS1538: Introduction to Simultions Some Well-Known Proility Distriutions Bernoulli Binomil Geometric Negtive Binomil Poisson Uniform Exponentil Gmm Erlng Gussin/Norml Relevnce

### Mat 210 Updated on April 28, 2013

Mt Brief Clculus Mt Updted on April 8, Alger: m n / / m n m n / mn n m n m n n ( ) ( )( ) n terms n n n n n n ( )( ) Common denomintor: ( ) ( )( ) ( )( ) ( )( ) ( )( ) Prctice prolems: Simplify using common

### MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1

MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]

### MATH 144: Business Calculus Final Review

MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

### ES.182A Topic 32 Notes Jeremy Orloff

ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

### MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

### PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by

PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the

### Chapter 5 : Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll

### 7 - Continuous random variables

7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

### INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

### Continuous Random Variable X:

Continuous Rndom Vrile : The continuous rndom vrile hs its vlues in n intervl, nd it hs proility distriution unction or proility density unction p.d. stisies:, 0 & d Which does men tht the totl re under