STAT 450: Statistical Theory. Distribution Theory. Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6.
|
|
- Solomon Kelly
- 5 years ago
- Views:
Transcription
1 STAT 45: Statistical Theory Distribution Theory Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6. Basic Problem: Start with assumptions about f or CDF of random vector X (X 1,..., X p ). Define Y g(x 1,..., X p ) to be some function of X (usually some statistic of interest like a sample mean or correlation coefficient or a test statistic). How can we compute the distribution or CDF or density of Y? Univariate Techniques In a univariate problem we have X a real-valued random variable and Y, a real valued transformation of X. Often we start with X and are told either the density or CDF of X and then asked to find the distribution of something like X 2 or sin(x) a function of X which we call a transformation. I am going to show you the steps you should follow in an example problem. Method 1: compute the CDF by integration and differentiate to find f Y. Example: Suppose U Uniform[, 1]. What is the distribution of log(x)?. Step 1: Make sure you give the transformed variable a name. Let Y log U. Step 2: Write down the definition of the CDF of Y : F Y (y) P (Y y). Step 3: Substitute in the definition of Y in terms of the original variable in this case, U: F Y (y) P (Y y) P ( log(u) y). Step 4: Solve the inequality to get, hopefully, one or more intervals for U. In this case log(u) y is the same as log(u) y 1
2 which is the same as So U e y. P ( log(u) Y ) P ( U e y). Step 5: now use the assumptions to write the resulting probability in terms of F U. P ( U e y) 1 P ( U < e y) 1 F U ( e y ). Step 6: Differentiate the resulting formula with respect to y. In this case the formula for F U (u) needs to be written carefully: 1 u > 1 F U (u) u < u < 1 u The quantity e y is never negative but if y < e y > 1 so { 1 F U (e y 1 y < ) 1 e y y Now we can differentiate. For negative y the derivative is. For positive y the derivative is e y. At y the derivative does not exist and the density f Y (y) does not have a unique definition So our density is y < f Y (y) undefined y e y y > This is the standard exponential density so we would finish the problem by saying that log(u) has a standard exponential distribution. I have done the individual steps in a bit of detail but here are the crucial pieces again in a chain of equalities: F Y (y) P (Y y) P ( log U y) P (log U y) P (U e y ) { 1 e y y > y. 2
3 Differentiating we find f Y (y) d dy F Y (y) so Y has standard exponential distribution. Example: Z N(, 1), i.e. and Y Z 2. Then Now differentiate y < undefined y e y y > f Z (z) 1 2π e z2 /2 F Y (y) P (Z 2 y) { y < P ( y Z y) y. P ( y Z y) F Z ( y) F Z ( y) to get Then f Y (y) [ y < d FZ ( y) F dy Z ( y) ] y > undefined y. d dy F Z( y) f Z ( y) d dy 1 2π exp y ( ( y) 2 /2 ) 1 2 y 1/ πy e y/2. (Similar formula for other derivative.) Thus 1 2πy e y/2 y > f Y (y) y < undefined y. 3
4 This is the density of a χ 2 1 random variable because the chi-squared distributing with ν degrees of freedom is defined to be the distribution of the sum of ν squared independent standard normal variables. I am now going to rewrite this density using indicator notation which can be very useful: { 1 y > 1(y > ) y which we use to write f Y (y) 1 2πy e y/2 1(y > ) (changing definition unimportantly at y ). Notice: I never evaluated F Y before differentiating it. In fact F Y and F Z are integrals I can t do but I can differentiate them anyway. Remember the fundamental theorem of calculus: d dx x a f(y) dy f(x) at any x where f is continuous. Summary: for Y g(x) with X and Y each real valued P (Y y) P (g(x) y) Take d/dy to compute the density f Y (y) d dy P (X g 1 (, y]). {x:g(x) y} Often can differentiate without doing integral. f X (x) dx. Method 2: Change of variables. Assume g is one to one. I do: g is increasing and differentiable. Interpretation of density (based on density F ): f Y (y) P (y Y y + δy) lim δy δy F Y (y + δy) F Y (y) lim δy δy 4
5 and P (x X x + δx) f X (x) lim. δx δx Now assume y g(x). Define δy by y + δy g(x + δx). Then Get Take limit to get or P (y Y g(x + δx)) P (x X x + δx). P (y Y y + δy)) δy P (x X x + δx)/δx {g(x + δx) y}/δx. f Y (y) f X (x)/g (x) f Y (g(x))g (x) f X (x). Alternative view: Each probability is integral of a density. The first is the integral of the density of Y over the small interval from y g(x) to y g(x + δx). The interval is narrow so f Y is nearly constant and P (y Y g(x + δx)) f Y (y)(g(x + δx) g(x)). Since g has a derivative the difference and we get g(x + δx) g(x) δxg (x) P (y Y g(x + δx)) f Y (y)g (x)δx. Same idea applied to P (x X x + δx) gives so that or, cancelling the δx in the limit P (x X x + δx) f X (x)δx f Y (y)g (x)δx f X (x)δx f Y (y)g (x) f X (x). If you remember y g(x) then you get f X (x) f Y (g(x))g (x). 5
6 Or solve y g(x) to get x in terms of y, that is, x g 1 (y) and then f Y (y) f X (g 1 (y))/g (g 1 (y)). This is just the change of variables formula for doing integrals. Remark: For g decreasing g < but Then the interval (g(x), g(x + δx)) is really (g(x + δx), g(x)) so that g(x) g(x + δx) g (x)δx. In both cases this amounts to the formula f X (x) f Y (g(x)) g (x). Mnemonic: f Y (y)dy f X (x)dx. Example: : X Weibull(shape α, scale β) or f X (x) α β ( ) α 1 x exp { (x/β) α } 1(x > ). β Let Y log X or g(x) log(x). Solve y log x: x exp(y) or g 1 (y) e y. Then g (x) 1/x and 1/g (g 1 (y)) 1/(1/e y ) e y. Hence ( ) e y α 1 exp { (e y /β) α } 1(e y > )e y. f Y (y) α β β For any y, e y > so indicator 1. So Define φ log β and θ 1/α; then, f Y (y) α β α exp {αy eαy /β α }. f Y (y) 1 θ exp { y φ θ { }} y φ exp. θ Extreme Value density with location parameter φ and scale parameter θ. (Note: several distributions are called Extreme Value.) Marginalization 6
7 Simplest multivariate problem: (or in general Y is any X j ). X (X 1,..., X p ), Y X 1 Theorem 1 If X has density f(x 1,..., x p ) and q < p then Y (X 1,..., X q ) has density f Y (x 1,..., x q ) f(x 1,..., x p ) dx q+1... dx p. f X1,...,X q is the marginal density of X 1,..., X q and f X the joint density of X but they are both just densities. Marginal just to distinguish from the joint density of X. Example: : The function is a density provided The integral is f(x 1, x 2 ) Kx 1 x 2 1(x 1 >, x 2 >, x 1 + x 2 < 1) P (X R 2 ) K 1 1 x1 f(x 1, x 2 ) dx 1 dx 2 1. x 1 x 2 dx 1 dx 2 K so K 24. The marginal density of x 1 is f X1 (x 1 ) 24 24x 1 x 2 1 x 1 (1 x 1 ) 2 dx 1 /2 K(1/2 2/3 + 1/4)/2 K/24 1(x 1 >, x 2 >, x 1 + x 2 < 1) dx 2 1 x1 x 1 x 2 1( < x 1 < 1)dx 2 12x 1 (1 x 1 ) 2 1( < x 1 < 1). 7
8 This is a Beta(2, 3) density. General problem has Y (Y 1,..., Y q ) with Y i g i (X 1,..., X p ). Case 1: q > p. Y won t have density for smooth g. Y will have a singular or discrete distribution. Problem rarely of real interest. (But, e.g., residuals have singular distribution.) Case 2: q p. We use a change of variables formula which generalizes the one derived above for the case p q 1. (See below.) Case 3: q < p. Pad out Y add on p q more variables (carefully chosen) say Y q+1,..., Y p. Find functions g q+1,..., g p. Define for q < i p, Y i g i (X 1,..., X p ) and Z (Y 1,..., Y p ). Choose g i so that we can use change of variables on g (g 1,..., g p ) to compute f Z. Find f Y by integration: f Y (y 1,..., y q ) f Z (y 1,..., y q, z q+1,..., z p )dz q+1... dz p Change of Variables Suppose Y g(x) R p with X R p having density f X. Assume g is a one to one ( injective ) map, i.e., g(x 1 ) g(x 2 ) if and only if x 1 x 2. Find f Y : Step 1: Solve for x in terms of y: x g 1 (y). Step 2: Use basic equation: and rewrite it in the form Interpretation of derivative dx dy f Y (y)dy f X (x)dx f Y (y) f X (g 1 (y)) dx dy. when p > 1: dx dy which is the so called Jacobian. ( ) det xi y j 8
9 Equivalent formula inverts the matrix: This notation means dy dx det f Y (y) f X(g 1 (y)) dy dx y 1 x 1 y 1 x 2 y p x 1. y p x 2 but with x replaced by the corresponding value of y, that is, replace x by g 1 (y). y 1 x p y p x p Example: : The density f X (x 1, x 2 ) 1 { } 2π exp x2 1 + x is the standard bivariate normal density. Let Y (Y 1, Y 2 ) where Y 1 X X 2 2 and Y 2 < 2π is angle from the positive x axis to the ray from the origin to the point (X 1, X 2 ). I.e., Y is X in polar co-ordinates. Solve for x in terms of y: so that X 1 Y 1 cos(y 2 ) X 2 Y 1 sin(y 2 ) g(x 1, x 2 ) (g 1 (x 1, x 2 ), g 2 (x 1, x 2 )) ( x x 2 2, argument(x 1, x 2 )) g 1 (y 1, y 2 ) (g1 1 (y 1, y 2 ), g2 1 (y 1, y 2 )) (y 1 cos(y 2 ), y 1 sin(y 2 )) ( dx dy det cos(y2 ) y 1 sin(y 2 ) sin(y 2 ) y 1 cos(y 2 ) y 1. It follows that f Y (y 1, y 2 ) 1 { } 2π exp y2 1 y 1 1( y 1 < )1( y 2 < 2π). 2 9 )
10 Next: marginal densities of Y 1, Y 2? Factor f Y as f Y (y 1, y 2 ) h 1 (y 1 )h 2 (y 2 ) where h 1 (y 1 ) y 1 e y2 1 /2 1( y 1 < ) and Then h 2 (y 2 ) 1( y 2 < 2π)/(2π). f Y1 (y 1 ) h 1 (y 1 ) h 1 (y 1 )h 2 (y 2 ) dy 2 h 2 (y 2 ) dy 2 so marginal density of Y 1 is a multiple of h 1. Multiplier makes f Y1 1 but in this case so that h 2 (y 2 ) dy 2 2π (2π) 1 dy 2 1 f Y1 (y 1 ) y 1 e y2 1 /2 1( y 1 < ). (Special case of Weibull or Rayleigh distribution.) Similarly f Y2 (y 2 ) 1( y 2 < 2π)/(2π) which is the Uniform((, 2π) density. Exercise: W Y1 2 /2 has standard exponential distribution. Recall: by definition U Y1 2 has a χ 2 distribution on 2 degrees of freedom. Exercise: find χ 2 2 density. Note: We show below factorization of density is equivalent to independence. 1
STAT 801: Mathematical Statistics. Distribution Theory
STAT 81: Mathematical Statistics Distribution Theory Basic Problem: Start with assumptions about f or CDF of random vector X (X 1,..., X p ). Define Y g(x 1,..., X p ) to be some function of X (usually
More informationSTAT 450: Statistical Theory. Distribution Theory. Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6.
STAT 450: Statistical Theory Distribution Theory Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6. Example: Why does t-statistic have t distribution? Ingredients: Sample X 1,...,X n from
More information2 Functions of random variables
2 Functions of random variables A basic statistical model for sample data is a collection of random variables X 1,..., X n. The data are summarised in terms of certain sample statistics, calculated as
More informationChange Of Variable Theorem: Multiple Dimensions
Change Of Variable Theorem: Multiple Dimensions Moulinath Banerjee University of Michigan August 30, 01 Let (X, Y ) be a two-dimensional continuous random vector. Thus P (X = x, Y = y) = 0 for all (x,
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationSTAT 450. Moment Generating Functions
STAT 450 Moment Generating Functions There are many uses of generating functions in mathematics. We often study the properties of a sequence a n of numbers by creating the function a n s n n0 In statistics
More informationMath 3215 Intro. Probability & Statistics Summer 14. Homework 5: Due 7/3/14
Math 325 Intro. Probability & Statistics Summer Homework 5: Due 7/3/. Let X and Y be continuous random variables with joint/marginal p.d.f. s f(x, y) 2, x y, f (x) 2( x), x, f 2 (y) 2y, y. Find the conditional
More informationMTH739U/P: Topics in Scientific Computing Autumn 2016 Week 6
MTH739U/P: Topics in Scientific Computing Autumn 16 Week 6 4.5 Generic algorithms for non-uniform variates We have seen that sampling from a uniform distribution in [, 1] is a relatively straightforward
More informationChapter 2: Differentiation
Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationChapter 2: Differentiation
Chapter 2: Differentiation Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 82 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationMultiple Random Variables
Multiple Random Variables Joint Probability Density Let X and Y be two random variables. Their joint distribution function is F ( XY x, y) P X x Y y. F XY ( ) 1, < x
More informationDefinition: If y = f(x), then. f(x + x) f(x) y = f (x) = lim. Rules and formulas: 1. If f(x) = C (a constant function), then f (x) = 0.
Definition: If y = f(x), then Rules and formulas: y = f (x) = lim x 0 f(x + x) f(x). x 1. If f(x) = C (a constant function), then f (x) = 0. 2. If f(x) = x k (a power function), then f (x) = kx k 1. 3.
More informationTransformation of Probability Densities
Transformation of Probability Densities This Wikibook shows how to transform the probability density of a continuous random variable in both the one-dimensional and multidimensional case. In other words,
More informationHigher Mathematics Course Notes
Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that
More informationDistributions of Functions of Random Variables. 5.1 Functions of One Random Variable
Distributions of Functions of Random Variables 5.1 Functions of One Random Variable 5.2 Transformations of Two Random Variables 5.3 Several Random Variables 5.4 The Moment-Generating Function Technique
More informationChapter 4 Multiple Random Variables
Review for the previous lecture Theorems and Examples: How to obtain the pmf (pdf) of U = g ( X Y 1 ) and V = g ( X Y) Chapter 4 Multiple Random Variables Chapter 43 Bivariate Transformations Continuous
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More informationMath 5a Reading Assignments for Sections
Math 5a Reading Assignments for Sections 4.1 4.5 Due Dates for Reading Assignments Note: There will be a very short online reading quiz (WebWork) on each reading assignment due one hour before class on
More informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationTangent Planes, Linear Approximations and Differentiability
Jim Lambers MAT 80 Spring Semester 009-10 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationThis exam is closed book and closed notes. (You will have access to a copy of the Table of Common Distributions given in the back of the text.
TEST #3 STA 5326 December 4, 214 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. (You will have access to
More informationconditional cdf, conditional pdf, total probability theorem?
6 Multiple Random Variables 6.0 INTRODUCTION scalar vs. random variable cdf, pdf transformation of a random variable conditional cdf, conditional pdf, total probability theorem expectation of a random
More information6.6 Inverse Trigonometric Functions
6.6 6.6 Inverse Trigonometric Functions We recall the following definitions from trigonometry. If we restrict the sine function, say fx) sinx, π x π then we obtain a one-to-one function. π/, /) π/ π/ Since
More informationFunctions of Random Variables
Functions of Random Variables Statistics 110 Summer 2006 Copyright c 2006 by Mark E. Irwin Functions of Random Variables As we ve seen before, if X N(µ, σ 2 ), then Y = ax + b is also normally distributed.
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01: Probability and Statistics for Engineers Spring 2013 Contents 1 Joint Probability Distributions 2 1.1 Two Discrete
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.13 INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) by A.J.Hobson 13.13.1 Introduction 13.13. The second moment of a volume of revolution about the y-axis 13.13.3
More informationMATH 250 TOPIC 13 INTEGRATION. 13B. Constant, Sum, and Difference Rules
Math 5 Integration Topic 3 Page MATH 5 TOPIC 3 INTEGRATION 3A. Integration of Common Functions Practice Problems 3B. Constant, Sum, and Difference Rules Practice Problems 3C. Substitution Practice Problems
More informationMultivariable Calculus and Matrix Algebra-Summer 2017
Multivariable Calculus and Matrix Algebra-Summer 017 Homework 4 Solutions Note that the solutions below are for the latest version of the problems posted. For those of you who worked on an earlier version
More information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
More information13 Implicit Differentiation
- 13 Implicit Differentiation This sections highlights the difference between explicit and implicit expressions, and focuses on the differentiation of the latter, which can be a very useful tool in mathematics.
More informationB553 Lecture 1: Calculus Review
B553 Lecture 1: Calculus Review Kris Hauser January 10, 2012 This course requires a familiarity with basic calculus, some multivariate calculus, linear algebra, and some basic notions of metric topology.
More informationPartial Derivatives Formulas. KristaKingMath.com
Partial Derivatives Formulas KristaKingMath.com Domain and range of a multivariable function A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D
More informationLecture 13 - Wednesday April 29th
Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131
More informationThe Multivariate Normal Distribution 1
The Multivariate Normal Distribution 1 STA 302 Fall 2017 1 See last slide for copyright information. 1 / 40 Overview 1 Moment-generating Functions 2 Definition 3 Properties 4 χ 2 and t distributions 2
More informationThis does not cover everything on the final. Look at the posted practice problems for other topics.
Class 7: Review Problems for Final Exam 8.5 Spring 7 This does not cover everything on the final. Look at the posted practice problems for other topics. To save time in class: set up, but do not carry
More informationMath/Stats 425, Sec. 1, Fall 04: Introduction to Probability. Final Exam: Solutions
Math/Stats 45, Sec., Fall 4: Introduction to Probability Final Exam: Solutions. In a game, a contestant is shown two identical envelopes containing money. The contestant does not know how much money is
More informationReview for the First Midterm Exam
Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers
More information1 Solution to Problem 2.1
Solution to Problem 2. I incorrectly worked this exercise instead of 2.2, so I decided to include the solution anyway. a) We have X Y /3, which is a - function. It maps the interval, ) where X lives) onto
More informationMSM120 1M1 First year mathematics for civil engineers Revision notes 4
MSM10 1M1 First year mathematics for civil engineers Revision notes 4 Professor Robert A. Wilson Autumn 001 Series A series is just an extended sum, where we may want to add up infinitely many numbers.
More informationOrder Statistics and Distributions
Order Statistics and Distributions 1 Some Preliminary Comments and Ideas In this section we consider a random sample X 1, X 2,..., X n common continuous distribution function F and probability density
More informationp. 6-1 Continuous Random Variables p. 6-2
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability (>). Often, there is interest in random variables
More informationStat 5101 Lecture Slides: Deck 8 Dirichlet Distribution. Charles J. Geyer School of Statistics University of Minnesota
Stat 5101 Lecture Slides: Deck 8 Dirichlet Distribution Charles J. Geyer School of Statistics University of Minnesota 1 The Dirichlet Distribution The Dirichlet Distribution is to the beta distribution
More informationLecture 3: Random variables, distributions, and transformations
Lecture 3: Random variables, distributions, and transformations Definition 1.4.1. A random variable X is a function from S into a subset of R such that for any Borel set B R {X B} = {ω S : X(ω) B} is an
More informationMath 101 Fall 2006 Exam 1 Solutions Instructor: S. Cautis/M. Simpson/R. Stong Thursday, October 5, 2006
Math 101 Fall 2006 Exam 1 Solutions Instructor: S. Cautis/M. Simpson/R. Stong Thursday, October 5, 2006 Instructions: This is a closed book, closed notes exam. Use of calculators is not permitted. You
More information4. CONTINUOUS RANDOM VARIABLES
IA Probability Lent Term 4 CONTINUOUS RANDOM VARIABLES 4 Introduction Up to now we have restricted consideration to sample spaces Ω which are finite, or countable; we will now relax that assumption We
More informationPartial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.
Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,
More informationExample. Evaluate. 3x 2 4 x dx.
3x 2 4 x 3 + 4 dx. Solution: We need a new technique to integrate this function. Notice that if we let u x 3 + 4, and we compute the differential du of u, we get: du 3x 2 dx Going back to our integral,
More informationThe Chain Rule. The Chain Rule. dy dy du dx du dx. For y = f (u) and u = g (x)
AP Calculus Mrs. Jo Brooks The Chain Rule To find the derivative of more complicated functions, we use something called the chain rule. It can be confusing unless you keep yourself organized as you go
More informationf(x, y) = 1 sin(2x) sin(2y) and determine whether each is a maximum, minimum, or saddle. Solution: Critical points occur when f(x, y) = 0.
Some Answers to Exam Name: (4) Find all critical points in [, π) [, π) of the function: f(x, y) = sin(x) sin(y) 4 and determine whether each is a maximum, minimum, or saddle. Solution: Critical points
More informationContinuous Random Variables
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random variables
More informationComplex Differentials and the Stokes, Goursat and Cauchy Theorems
Complex Differentials and the Stokes, Goursat and Cauchy Theorems Benjamin McKay June 21, 2001 1 Stokes theorem Theorem 1 (Stokes) f(x, y) dx + g(x, y) dy = U ( g y f ) dx dy x where U is a region of the
More informationTransformations and Expectations
Transformations and Expectations 1 Distributions of Functions of a Random Variable If is a random variable with cdf F (x), then any function of, say g(), is also a random variable. Sine Y = g() is a function
More informationCHAPTER 2. The Derivative
CHAPTER The Derivative Section.1 Tangent Lines and Rates of Change Suggested Time Allocation: 1 to 1 1 lectures Slides Available on the Instructor Companion Site for this Text: Figures.1.1,.1.,.1.7,.1.11
More informationJUST THE MATHS UNIT NUMBER NUMERICAL MATHEMATICS 6 (Numerical solution) of (ordinary differential equations (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 17.6 NUMERICAL MATHEMATICS 6 (Numerical solution) of (ordinary differential equations (A)) by A.J.Hobson 17.6.1 Euler s unmodified method 17.6.2 Euler s modified method 17.6.3
More informationMA2AA1 (ODE s): The inverse and implicit function theorem
MA2AA1 (ODE s): The inverse and implicit function theorem Sebastian van Strien (Imperial College) February 3, 2013 Differential Equations MA2AA1 Sebastian van Strien (Imperial College) 0 Some of you did
More information4. Distributions of Functions of Random Variables
4. Distributions of Functions of Random Variables Setup: Consider as given the joint distribution of X 1,..., X n (i.e. consider as given f X1,...,X n and F X1,...,X n ) Consider k functions g 1 : R n
More informationMath 127C, Spring 2006 Final Exam Solutions. x 2 ), g(y 1, y 2 ) = ( y 1 y 2, y1 2 + y2) 2. (g f) (0) = g (f(0))f (0).
Math 27C, Spring 26 Final Exam Solutions. Define f : R 2 R 2 and g : R 2 R 2 by f(x, x 2 (sin x 2 x, e x x 2, g(y, y 2 ( y y 2, y 2 + y2 2. Use the chain rule to compute the matrix of (g f (,. By the chain
More informationThere are some trigonometric identities given on the last page.
MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during
More informationCalculus I Practice Exam 2
Calculus I Practice Exam 2 Instructions: The exam is closed book, closed notes, although you may use a note sheet as in the previous exam. A calculator is allowed, but you must show all of your work. Your
More informationSTA 2201/442 Assignment 2
STA 2201/442 Assignment 2 1. This is about how to simulate from a continuous univariate distribution. Let the random variable X have a continuous distribution with density f X (x) and cumulative distribution
More informationSec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx
Math 2204 Multivariable Calc Chapter 14: Partial Derivatives I. Review from math 1225 A. First Derivative Sec. 14.3: Partial Derivatives 1. Def n : The derivative of the function f with respect to the
More informationEssential Maths 1. Macquarie University MAFC_Essential_Maths Page 1 of These notes were prepared by Anne Cooper and Catriona March.
Essential Maths 1 The information in this document is the minimum assumed knowledge for students undertaking the Macquarie University Masters of Applied Finance, Graduate Diploma of Applied Finance, and
More informationMATH 307: Problem Set #3 Solutions
: Problem Set #3 Solutions Due on: May 3, 2015 Problem 1 Autonomous Equations Recall that an equilibrium solution of an autonomous equation is called stable if solutions lying on both sides of it tend
More informationCore Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
More informationSchool of the Art Institute of Chicago. Calculus. Frank Timmes. flash.uchicago.edu/~fxt/class_pages/class_calc.
School of the Art Institute of Chicago Calculus Frank Timmes ftimmes@artic.edu flash.uchicago.edu/~fxt/class_pages/class_calc.shtml Syllabus 1 Aug 29 Pre-calculus 2 Sept 05 Rates and areas 3 Sept 12 Trapezoids
More informationChapter 2 Continuous Distributions
Chapter Continuous Distributions Continuous random variables For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following
More informationP1 Calculus II. Partial Differentiation & Multiple Integration. Prof David Murray. dwm/courses/1pd
P1 2017 1 / 39 P1 Calculus II Partial Differentiation & Multiple Integration Prof David Murray david.murray@eng.ox.ac.uk www.robots.ox.ac.uk/ dwm/courses/1pd 4 lectures, MT 2017 P1 2017 2 / 39 Motivation
More information0.0.1 Moment Generating Functions
0.0.1 Moment Generating Functions There are many uses of generating functions in mathematics. We often study the properties of a sequence a n of numbers by creating the function a n s n n0 In statistics
More informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
More informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More informationHandout 5, Summer 2014 Math May Consider the following table of values: x f(x) g(x) f (x) g (x)
Handout 5, Summer 204 Math 823-7 29 May 204. Consider the following table of values: x f(x) g(x) f (x) g (x) 3 4 8 4 3 4 2 9 8 8 3 9 4 Let h(x) = (f g)(x) and l(x) = g(f(x)). Compute h (3), h (4), l (8),
More informationMultivariate distributions
CHAPTER Multivariate distributions.. Introduction We want to discuss collections of random variables (X, X,..., X n ), which are known as random vectors. In the discrete case, we can define the density
More informationAnalysis/Calculus Review Day 2
Analysis/Calculus Review Day 2 Arvind Saibaba arvindks@stanford.edu Institute of Computational and Mathematical Engineering Stanford University September 14, 2010 Limit Definition Let A R, f : A R and
More informationNumerical differentiation
Numerical differentiation Paul Seidel 1801 Lecture Notes Fall 011 Suppose that we have a function f(x) which is not given by a formula but as a result of some measurement or simulation (computer experiment)
More informationLecture 1: August 28
36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 1: August 28 Our broad goal for the first few lectures is to try to understand the behaviour of sums of independent random
More information1 Lecture 39: The substitution rule.
Lecture 39: The substitution rule. Recall the chain rule and restate as the substitution rule. u-substitution, bookkeeping for integrals. Definite integrals, changing limits. Symmetry-integrating even
More information(c) The first thing to do for this problem is to create a parametric curve for C. One choice would be. (cos(t), sin(t)) with 0 t 2π
1. Let g(x, y) = (y, x) ompute gds for a circle with radius 1 centered at the origin using the line integral. (Hint: use polar coordinates for your parametrization). (a) Write out f((t)) so that f is a
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationAppendix 5. Derivatives of Vectors and Vector-Valued Functions. Draft Version 24 November 2008
Appendix 5 Derivatives of Vectors and Vector-Valued Functions Draft Version 24 November 2008 Quantitative genetics often deals with vector-valued functions and here we provide a brief review of the calculus
More informationSecond Order ODEs. Second Order ODEs. In general second order ODEs contain terms involving y, dy But here only consider equations of the form
Second Order ODEs Second Order ODEs In general second order ODEs contain terms involving y, dy But here only consider equations of the form A d2 y dx 2 + B dy dx + Cy = 0 dx, d2 y dx 2 and F(x). where
More informationThe Multivariate Normal Distribution. In this case according to our theorem
The Multivariate Normal Distribution Defn: Z R 1 N(0, 1) iff f Z (z) = 1 2π e z2 /2. Defn: Z R p MV N p (0, I) if and only if Z = (Z 1,..., Z p ) T with the Z i independent and each Z i N(0, 1). In this
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationSystems of Linear ODEs
P a g e 1 Systems of Linear ODEs Systems of ordinary differential equations can be solved in much the same way as discrete dynamical systems if the differential equations are linear. We will focus here
More informationMATH 411 NOTES (UNDER CONSTRUCTION)
MATH 411 NOTES (NDE CONSTCTION 1. Notes on compact sets. This is similar to ideas you learned in Math 410, except open sets had not yet been defined. Definition 1.1. K n is compact if for every covering
More informationSolution. This is a routine application of the chain rule.
EXAM 2 SOLUTIONS 1. If z = e r cos θ, r = st, θ = s 2 + t 2, find the partial derivatives dz ds chain rule. Write your answers entirely in terms of s and t. dz and dt using the Solution. This is a routine
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Exercise 4.1 Let X be a random variable with p(x)
More informationCh3. Generating Random Variates with Non-Uniform Distributions
ST4231, Semester I, 2003-2004 Ch3. Generating Random Variates with Non-Uniform Distributions This chapter mainly focuses on methods for generating non-uniform random numbers based on the built-in standard
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationCalculus II Practice Test Problems for Chapter 7 Page 1 of 6
Calculus II Practice Test Problems for Chapter 7 Page of 6 This is a set of practice test problems for Chapter 7. This is in no way an inclusive set of problems there can be other types of problems on
More informationMath Refresher Course
Math Refresher Course Columbia University Department of Political Science Fall 2007 Day 2 Prepared by Jessamyn Blau 6 Calculus CONT D 6.9 Antiderivatives and Integration Integration is the reverse of differentiation.
More informationf(x) g(x) = [f (x)g(x) dx + f(x)g (x)dx
Chapter 7 is concerned with all the integrals that can t be evaluated with simple antidifferentiation. Chart of Integrals on Page 463 7.1 Integration by Parts Like with the Chain Rule substitutions with
More informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More informationIntroduction to Probability Theory
Introduction to Probability Theory Ping Yu Department of Economics University of Hong Kong Ping Yu (HKU) Probability 1 / 39 Foundations 1 Foundations 2 Random Variables 3 Expectation 4 Multivariate Random
More information4 Partial Differentiation
4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which
More informationProbability Models. 4. What is the definition of the expectation of a discrete random variable?
1 Probability Models The list of questions below is provided in order to help you to prepare for the test and exam. It reflects only the theoretical part of the course. You should expect the questions
More informationLecture 25: Review. Statistics 104. April 23, Colin Rundel
Lecture 25: Review Statistics 104 Colin Rundel April 23, 2012 Joint CDF F (x, y) = P [X x, Y y] = P [(X, Y ) lies south-west of the point (x, y)] Y (x,y) X Statistics 104 (Colin Rundel) Lecture 25 April
More information4. We accept without proofs that the following functions are differentiable: (e x ) = e x, sin x = cos x, cos x = sin x, log (x) = 1 sin x
4 We accept without proofs that the following functions are differentiable: (e x ) = e x, sin x = cos x, cos x = sin x, log (x) = 1 sin x x, x > 0 Since tan x = cos x, from the quotient rule, tan x = sin
More informationMath 115 Second Midterm March 25, 2010
Math 115 Second Midterm March 25, 2010 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover. There are 8 problems.
More informationStat751 / CSI771 Midterm October 15, 2015 Solutions, Comments. f(x) = 0 otherwise
Stat751 / CSI771 Midterm October 15, 2015 Solutions, Comments 1. 13pts Consider the beta distribution with PDF Γα+β ΓαΓβ xα 1 1 x β 1 0 x < 1 fx = 0 otherwise, for fixed constants 0 < α, β. Now, assume
More information