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### math-2011-note3

Course: ECON 205, Spring 2011
School: Korea University
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Word Count: 960

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Inverse 4.2 Functions Denition 4 (p.76). For a given function f : E1 R1 , E1 R1 , we say a function g : E2 R1 , E2 R1 , is an inverse of f if g( f (x)) = x for all x E1 and f (g(y)) = y for all y E2 . 1 Example: f (x) = 3 2x, g(y) = 2 (3 y) If f has an inverse we say that f is invertible, and its inverse function is written as f 1 . Ex Suppose f (x) = x2 for x &gt; 0. What is f 1 (x)? Figure 9:...

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Inverse 4.2 Functions Denition 4 (p.76). For a given function f : E1 R1 , E1 R1 , we say a function g : E2 R1 , E2 R1 , is an inverse of f if g( f (x)) = x for all x E1 and f (g(y)) = y for all y E2 . 1 Example: f (x) = 3 2x, g(y) = 2 (3 y) If f has an inverse we say that f is invertible, and its inverse function is written as f 1 . Ex Suppose f (x) = x2 for x > 0. What is f 1 (x)? Figure 9: Graph of y = x2 and its inverse function 4.3 The Derivative of the Inverse Function The derivative of the inverse function f 1 has a close relationship with the derivative of f . Theorem 8 (Thm 4.3; Inverse Function Theorem, p.79). Let f be a C1 function dened on the interval I R1 . If f (x) = 0 for all x I, then (a) f is invertible on I, (b) its inverse g is C1 function on the interval f (I ), and (c) for all y f (I ), g (y) = 1 . f (g(y)) 13 Proof. Since f is either increasing or decreasing on I (why?), f 1 is well dened on f (I ). For each y f (I ), we have f (g(y)) = y. By differentiating both sides with respect 1 to y, we obtain f (g(y)) g (y) = 1. Hence g (y) = f (g(y)) , and g (y) is continuous since f and g are continuous. m 4.4 The Derivative of x n Theorem 9 (Thm 4.4, p.80). For any positive integer n, d1 n dx x 1 = 1 x n 1 . n 1 Proof. The inverse of y = x n is x = yn . Hence, d1 1 1 1n 1 11 xn = = n1 = x n = x n 1 . 1 n 1 dx n n n(x n ) nx n Theorem 10 (Thm 4.5, p.81). For any positive integers m and n, m dm n dx x m = m x n 1 . n 1 Proof. Since x n = (x n )m , we can apply the Chain Rule; 1 dm 11 mm x n = m ( x n ) m 1 ( ) x n 1 = x n 1 . dx n n 5 Exponents and Logarithms (ch.5) 5.1 Exponential Functions A function whose variable x appears as an exponent is called an exponential function, for example f (x) = 2x. If x is a positive integer, 2x means multiply 2 by itself x times. If x = 0, 20 = 1 by denition. 1 1 If x = n , 2 n = n 2. m If x = m , 2 n = ( n 2)m . n If x is a negative number, 2x = 1 . 2|x| Exponential function ax is dened only for a > 0. 14 Figure 10: graph of an exponential function 5.2 The Number e If the annual interest rate is r, one can get A(1 + r) after one year from a bank deposit of amount A. r Suppose that the bank compounds interest 4 times a year, paying 4 principal as interest at the end of each quarter. r Then after one year the deposit becomes A(1 + 4 )4 . r If the bank compounds interest n times a year, the deposit A(1 becomes + n )n after one year. Suppose A = 1, r = 1, and consider n . The limit exists and it is 1 e lim (1 + )n 2.718... n n The function f (x) = ex is called the exponential function and sometimes written as exp(x). Theorem 11 (Thm 5.1, p.87). r lim (1 + )n = er . n n Proof. For r > 0, we have r 1 (n / r )r 1 n/r lim (1 + )n = lim (1 + ) = lim (1 + ) n n n n n/r n/r r = er . For r < 0, we have r |r | n |r | n n lim (1 + )n = lim (1 )n = lim = lim n n n n n |r | n n n | r | n = lim (1 + ) = e|r| = er . n n |r | 15 n 5.3 Logarithms Exponential function f (x) = ax with a > 1 is an increasing function. Thus it has an inverse function g(y) = loga y, called base a logarithm; x = loga y y = ax . By denition of inverse functions, one has aloga y = y and loga ax = x. One of the commonly used logarithms is base 10 logarithm, which is written with capital L; y =Log x 10y = x. Another is base e logarithm, called natural logarithm, which is written as ln; y = ln x ey = x. Figure 11: graphs of exponential and logarithmic functions 5.4 Properties of exp and log Exponential functions satisfy a r a s = a r +s a r = 1 / a r a r / a s = a r s (ar )s = ars a0 = 1 Logarithmic functions satisfy loga (u v) = loga u + loga v, 16 loga (1/v) = loga v, loga (u/v) = loga u loga v, loga uv = v loga u, loga 1 = 0 5.5 Derivative of exp and log Theorem 12 (Thm 5.2, p.93). dx d 1 e = ex , ln x = dx dx x Proof. ln(x + h) ln x 1 x + h 1/x = ln = ln 1 + h h x 1/h Let m = 1/h. As h 0, m . By Theorem (5.1), we obtain 1/x ln(x + h) ln x = lim ln 1 + m h0 h m 1 = ln e1/x = . x x By differentiating both sides of ln e = x, one obtains 1d x d e = 1. Hence, ex = ex . x dx e dx 1/h . m lim Ex. Prove that dx dx b = bx ln b. (Hint: Let k = lnb.) 5.6 Applications to Economics (ch.3) Demand Function and Elasticity A demand function D( p) assigns the quantity demanded to each values of market price p. If we denote the quantity demanded by q, then q = D( p). Price elasticity of demand measures how sensitively the demand changes as price changes. It is dened as the ratio of percentage change in demand with respect to the percentage change in price. Formally, q / q q p p = lim = D ( p) p p/ p p p q q = lim One can also write = d ln q d ln p , since d ln q ln D( p + h) ln D( p) d ln q dq 1 1 1 = lim = = D ( p) 1 d ln p d ln p h0 ln( p + h) ln p dq d p q p dp 17
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Korea University - ECON - 205
6 Integration (A.4)6.1 Indenite Integral Consider a continuous function f (x), where f (x) &gt; 0 for all x. Consider the area under the graph of y = f (x) from a certain point a to anotherpoint x and denote it by A(x; a). What is the derivative of A(x;
Korea University - ECON - 205
9 Limits and Open Sets (ch.12)9.1 Sequences in Real NumbersA sequence of real numbers cfw_x1 , x2 , , xn , is an assignment of a real number xnto each natural number n.Denition 7 (p.255). Let cfw_x1 , x2 , , xn , be a sequence of real numbers. A rea
Korea University - ECON - 205
Theorem 34 (Thm 13.3, p.291). The general quadratic form Q(x1 , , xk ) = i j ai j xi x jcan be written as1a11 2 a12 1 a1kx12 1 a12 a22 1 a2k x2 22x1 x2 xk ... . ,.... . .....11xkakk2 a1 k2 a2 k or xT Ax, where A is a symmetric m
Korea University - ECON - 205
12 Calculus of Several Variables (ch.14)12.1 Partial DerivativeDenition 25 (p.300). Let F : Rk R. The partial derivative of f with respect to xi atx0 = (x0 , , x0 ) is dened as1kF (x0 , , x0 + h, , x0 ) F (x0 , , x0 , , x0 )F 0ii11kk(x ) = l
Korea University - ECON - 205
Theorem 41 (Thm 30.5, p.828). Let f : R1 R1 be a C2 function. For any point a &lt;b R1 , there is a point c (a, b) such that f (b) = f (a) + f (a)(b a) + 1 f (c)(b 2a)2 .Proof. Dene2f (b) f (a) f (a)(b a) .(b a)2Suppose that f (x) = M for all x (a, b
Korea University - ECON - 205
Example 6. Suppose that we have a production function Q = kxa yb . Then,Q= akxa1 yb ,x 2Q= abkxa1 yb1 , x yQ= bkxa yb1y 2Q= abkxa1 yb1 y x14 Some Linear Algebra14.1 Deniteness of Quadratic Forms (16.2)Denition 27. Let A be an m m symmetric
Korea University - ECON - 205
Theorem 53 (Thm 11.2, p.243). Let A = (a1 , a2 , , am). If a1 , a2 , , am are linearlyindependent if and only if det A = 0.Proof. a) only if: If a1 , a2 , , am are linearly independent, then A is one-to-one andonto, hence has an inverse function. Call
Korea University - ECON - 205
Denition 36 (p.161). An m m matrix A = (ai j ) is called an upper-triangular matrixif ai j = 0 for i &gt; j. A is called a lower-triangular matrix if ai j = 0 for i &lt; j. A is calleda diagonal matrix if ai j = 0 for i = j.Theorem 56 (Fact 26.11, p.731). Th
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #9This assignment is due at the beginning of class on Friday, November 30, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of exposit
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Covariance Zero Does Not Imply IndependenceThe following exercise illustrates that two random variables can have covariance zero yet need notbe independent.Exercise. Consider the random variable X dened by P (X = 1) = 1/4, P
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Fishers F -distribution and Statistical Hypothesis TestingFor Problem #4 on page 27 you are asked to manipulate Fishers F -distribution. The F -distributionarises in the following context. (See Section 10.9 in the Stat 251 tex
University of Regina - STAT - 351
Statistics 351.001 Fall 2008Final Exam InformationThe date and location of the nal exam are Friday, December 12, 2008, 9:00 a.m. 12:00 p.m. Education Building, room 230 (ED 230)Please ensure that you know where the location of the test room is and th
University of Regina - STAT - 351
Statistics 351 (Fall 2007)The Gamma FunctionSuppose that p &gt; 0, and deneup1 eu du.(p) :=0We call (p) the Gamma function and it appears in many of the formul of density functionsfor continuous random variables such as the Gamma distribution, Beta di
University of Regina - STAT - 351
Statistics 351 Fall 2007Independence of X and S 2Theorem. Suppose that X1 , . . . , Xn are independent N (0, 1) random variables. If1X=nnXii=11and S =n1n2(Xi X )2i=1denote the sample mean and sample variance, respectively, then X and S 2 a
University of Regina - STAT - 351
Statistics 351 (Fall 2007)The Jacobian for Polar CoordinatesExample. Determine the Jacobian for the change-of-variables from cartesian coordinatesto polar coordinates.Solution. The traditional letters to use arex = r cos and y = r sin .However, to a
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Review of Linear AlgebraSuppose that A is the symmetric matrix1 1 0A = 1 2 1 .013Determine the eigenvalues and eigenvectors of A.Recall that a real number is an eigenvalue of A if Av = v for some vector v = 0. We callv a
University of Regina - STAT - 351
Statistics 351 Midterm #1 October 10, 2007This exam has 4 problems and 6 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate yo
University of Regina - STAT - 351
Statistics 351 Midterm #1 October 6, 2008This exam has 4 problems and 6 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate you
University of Regina - STAT - 351
Statistics 351 Midterm #1 October 6, 2008This exam has 4 problems and 6 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate you
University of Regina - STAT - 351
Statistics 351 Midterm #2 November 16, 2007This exam is worth 50 points.There are 5 problems on 5 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in orde
University of Regina - STAT - 351
Statistics 351 (Fall 2007)The Density and Characteristic Function Denitions of Multivariate NormalitySuppose that the random vector X = (X, Y ) has a multivariate normal distribution with meanvector and covariance matrix given by=xyand =2xx y.
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #1 Solutions2. (a) If X Unif[0, 2], then FX (x) = x for 0 x 2, and if Y Exp(3), then FY (y ) = 1 ey/32for y &gt; 0. Since X and Y are independent, we conclude thatFX,Y (x, y ) = FX (x) FY (y ) =x1 ey/32for 0 x 2 and y &gt;
University of Regina - STAT - 351
Statistics 351 Fall 2007 Midterm #1 Solutions1. (a) By denition,x yfX (x) =2e exdy = 2ey= 2e2x , x &gt; 0,exandxyyx yfY (y ) =2e eydx = 2ex= 2ey 1 ey ,e0y &gt; 0.01. (b) Since fX,Y (x, y ) = fX (x) fY (y ), we immediately conclude that
University of Regina - STAT - 351
Statistics 351 Fall 2008 Midterm #1 Solutions1. (a) By denition,yefX (x) == ex ,ydy = exx &gt; 0.xNote that X Exp(1) so that E(X ) = 1.1. (b) By denition,yey dx = yey ,fY (y ) =y &gt; 0.0Note that Y (2, 1).1. (c) By denition,fY |X =x (y ) =
University of Regina - STAT - 351
Statistics 351 Fall 2008 Midterm #1 Solutions1. (a) By denition,xefY (y ) == ey ,xdx = eyy &gt; 0.yNote that Y Exp(1) so that E(Y ) = 1.1. (b) By denition,xex dy = xex ,fX (x) =x &gt; 0.0Note that X (2, 1).1. (c) By denition,fY |X =x (y ) =
University of Regina - STAT - 351
Statistics 351 Fall 2007 Midterm #2 Solutions1. (a) Let1 111B=and b =20so thatBX + b =1 111X1X2+20=X1 X2 2X1 + X2Y1Y2== Y.By Theorem V.3.1, we conclude that Y N (B + b, B B ) where1 111E(Y) = B + b =12+10=00andcov(Y) =
University of Regina - STAT - 351
Stat 351 Fall 2007Solutions to Assignment #4Problem #3, page 55: Suppose that X + Y = 2. By denition of conditional density,fX,X +Y (x, 2).fX +Y (2)fX |X +Y =2 (x) =We now nd the joint density fX,X +Y (x, 2). Let U = X and V = X + Y so that X = U a
University of Regina - STAT - 351
Stat 351 Fall 2007Solutions to Assignment #51. (a) We begin by calculating E(Y1 ). That is,E(Y1 ) = 1 P (Y = 1) + (1) P (Y = 1) = p (1 p) = 2p 1.We now notice that Sn+1 = Sn + Yn+1 . Therefore,E(Sn+1 |X1 , . . . , Xn ) = E(Sn + Yn+1 |X1 , . . . , Xn
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #7 Solutions1. If X1 , X2 are independent N (0, 1) random variables, then by Denition I, Y1 = X1 3X2 + 2 isnormal with mean E (Y1 ) = E (X1 ) 3E (X2 ) + 2 = 2 and variance var(Y1 ) = var(X1 3X2 + 2) =var(X1 ) + 9 var(X2 )
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #8 SolutionsExercise 7.1, page 134: By Denition I, we know that X and Y X are normally distributed.Therefore, by Theorem 7.1, X and Y X are independent if and only if cov(X, Y X ) = 0. Wecomputecov(X, Y X ) = cov(X, Y ) c
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #9 Solutions1. (a) By Denition I, we see that X1 X2 is normally distributed with meanE (X1 X2 ) = E (X1 ) E (X2 ) = 0and variancevar(X1 X2 ) = var(X1 ) + 2 var(X2 ) 2 cov(X1 , X2 ) = 1 + 2 22 = 1 2 .That is, X1 X2 = Y wh
University of Regina - STAT - 351
Statistics 351Intermediate ProbabilityFall 2008 (200830)Final Exam SolutionsInstructor: Michael Kozdron1. (a) We see that fX,Y (x, y ) 0 for all 0 &lt; x, y &lt; 1, and that1425y dy = y10xy dx dy =fX,Y (x, y ) dx dy =011y500= 1.0Thus, fX,Y i
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Simple Random WalkSuppose that Y1 , Y2 , . . . are i.i.d. random variables with P cfw_Y1 = 1 = P cfw_Y1 = 1 = 1/2,and dene the discrete time stochastic process cfw_Sn , n = 0, 1, . . . by setting S0 = 0 andnSn =Yi .i=1We
University of Regina - STAT - 351
Misprints and Corrections toAn Intermediate Course in ProbabilityMisprintsPage25272749516265656997124128140141141141141151156156183183183183184184184Line155111491391813314846945891212141481013Text
University of Regina - STAT - 351
Make sure that this examination has 13 numbered pagesUniversity of ReginaDepartment of Mathematics &amp; StatisticsFinal Examination200830(December 12, 2008)Statistics 351Intermediate ProbabilityName:Student Number:Instructor: Michael KozdronTime:
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #1This assignment is due at the beginning of class on Monday, September 10, 2007. You must submitall problems that are marked with an asterix (*).1.* Send me an email to say Hello. If I have never taught you before, tell
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #5This assignment is due at the beginning of class on Friday, October 5, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of expositio
University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
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University of Regina - STAT - 351
Statistics 351 Fall 2006 (Kozdron) Midterm #2 Solutions1. (a) Recall that a square matrix is strictly positive denite if and only if the determinantsof all of its upper block diagonal matrices are strictly positive. Since2 22 3=we see that det( 1 )
Lone Star College - BIOL - 2402
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St. Johns - PHI - 2240C
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