Random Variate Generation
|
|
- Margery Shepherd
- 5 years ago
- Views:
Transcription
1 CPSC 405 Random Variate Generation 2007W T1 Handout These notes present techniques to generate samples of desired probability distributions, and some fundamental results and techiques. Some of this material can be found in Chapter 8 (9 in 2nd ed.) of the book. 1 Inverse Transform Suppose we can generate a uniform random number r on [0 1]. How can we generate numbers x with a given pdf P (x)? To warm up our brain, let s first think about something else. Suppose we generate a uniform random number 0 < r < 1 and square it. So we have x = r 2. Clearly we also have 0 < x < 1. What is the pdf P (x) for x? A wild guess might be that it is just the square of the pdf for r, so x would also be uniform. It is however easy to see that this can t be true. Consider the probability p that x < 1/2. If x was uniform on [0 1] we would get p = 1/2. In order to get an x < 1/2 we must have gotten an r < 1/ 2. The probability for this is p = 1/ 2, not p = 1/2. So P (x) can t be uniform. To figure out what P (x) is, consider the probability p that x falls in the interval [x x + x]. In the limit x 0 we have, according to the definition of a pdf, p = P (x) x. So if we can figure out p, we can compute P (x). For x to fall in [x x + x], we must generate r in the range [ x x + x]. Since r is uniform, the probability for this (which is also p) is just the length of this interval. So we get p = x + x x, and we now use p = P (x) x and solve for P (x), obtaining x + x x P (x) =. x Taking the limit x 0 we thus obtain x + x x P (x) = lim x 0 x = d 1 x = dx 2 x. With this result, we now have an instance of the inverse transform method to generate random numbers with pdf 1/2 x: generate a uniform random number r and square it. The general form of the inverse transform method is obtained by computing the pdf of x = g(r) for some function g, and then trying to find a function g such that the desired pdf is obtained. Let us assume that g(r) is invertible with inverse g 1 (x). The chance that x lies in the interval [x x + dx] is P (x)dx, for infinitesimal dx. What values of r should we have gotten to get this? (Remember we are generating values x by calling a uniform random number generator to get r and then setting x = g(r).) We should have gotten an r in the interval [r r + dr], with r = g 1 (x) and r + dr = g 1 (x + dx). Write out the last formula and get r + dr = g 1 (x) + (g 1 ) (x)dx, 1
2 where the denotes the derivative. Using r = g 1 (x) we can simplify this to dr = (g 1 ) (x)dx. (1) The probability for the r value to be in the interval [r r + dr] is just dr. This is also the probability for x to be in [x x + dx], which is P (x)dx. Using Eq. 1, we get thus P (x)dx = (g 1 ) (x)dx, P (x) = (g 1 ) (x). Integrating both sides and remembering that F (x) = x P (y)dy, gives us or F (x) = g 1 (x), g(x) = F 1 (x), provided F (x) has an inverse. In summary, to generate a number x with pdf P (x) using the inverse transform method, we first figure out the cdf F (x) from P (x). We then invert that by solving r = F (x) for x, which gives the function F 1 (r). We then generate a uniform random number 0 < r < 1 and compute x = F 1 (r). 2 Pdf of a function of a random variable Suppose x has pdf P (x). What is the pdf Q(y) of y = g(x)? The chance for x to be in [x x + dx] is P (x)dx. Then y is in [y y + dy], with y = g(x) and dy = g (x)dx. The chance for y to be in that interval is per definition Q(y)dy. So we get, noting that g can be negative, Q(y) g (x) dx = P (x)dx, so Q(y) = P (x)/ g (x) = P (g 1 (y)) (g 1 ) (y). (2) Note that we have made an implicit assumption here that g(x) is monotone. If it is not, several intervals in x can map onto the same values y, and the inverse does not exist. This would complicate matters, and we shall not deal with this problem. If g(x) is monotone the derivative is either negative or positive (or zero), which means the derivative of the absolute value is the absolute value of the derivative, which was used in Eq. 2. Let s try it on a familiar example: P (x) is 1 on [0 1] (i.e., the uniform distribution) and g(x) = (b a)x + a. We know already that the result is a uniform pdf on [a b]. Inverting g(x) gives g 1 (y) = (y a)/(b a), and (g 1 ) (y) = 1/(b a), so Q(y) = 1/(b a) which is correct. 2
3 A linear transformation is often used to get a normal distribution with given µ and σ from the standard normal with µ = 1 and σ = 1, N(t) = 1 2π e t2 /2. If z denotes a standard normal variate, then x = µ + σz is normally distributed with those mean and variance as can be verified easily by using Eq Constructing the pdf from measured data In this section I will present an algorithm to generate samples from a continuous distribution for which we only know a finite number of measured data points. Suppose we have measured some parameter x in a system, and have recorded N + 1 values which we have sorted in increasing order x 1, x 2,..., x N+1. Our task is now to create samples from the unknown pdf P (x) underlying this data. There are several ways to do this and since we have only limited data we will have to make some guesses. Let us denote the N intervals by I k = [x k x k+1 ] for k = 1,..., N. We now want to assign equal probabilities to each of the intervals. Areas with small interval sizes will then have a higher P (x) as expected as there are more intervals per unit length, if all intervals are treated as equally probable. Let s begin with our old friend, the uniform random number generator on [0 1], and divide the unit interval into N equal intervals R k = [k (k + 1)]/N, with k = 1,..., N. The plan is now to generate r, figure out which interval R k it is in, look up the corresponding interval I k, and generate an appropriate value of x in that interval I k, depending on the location in the interval R k of r. If r is in the left of the interval R k a value from the left side of I k will be generated and the other way around. See Figure 1. Here is MATLAB code (empgen.m) that does it. It takes a vector data with a (sorted) data sample and returns a random number y generated from the empirical pdf. Note that the empirical pdf is never explicitly constructed. function y = empgen(data) N = length(data)-1; r = rand; % k is interval k = 1+floor(N*r); % relative offset in interval (0-1) offset = r*n - (k-1); % map to appropriate value of data y = data(k)*(1-offset) + data(k+1)*offset; 3
4 Figure 1: Mapping from the uniform random number r to the value x based on the measured data points. 4 Convolutions Let K y = r k, k=1 where r are taken from some (fixed) distribution P (r). The pdf of y, Q(y) is called a convolution of the distribution P (x). Note that it is just a sum, and a sum is the average up to a constant. There is no simple way to compute Q(y). It goes like this. Consider the K dimensional space R spanned by r k. Let < r k <, and P (r) is possibly zero on big regions on R. The equation K k=1 r k = y, for fixed y defines a hyperplane H in R. So Q(y) is just the probability density that the r k ly on the hyperplane H, which is Q(y) = P (r 1 )P (r 2 )... P (r K ))d K 1 r, (3) H which is a hypersurface integral. For example, consider K = 2 and take P (r) to be uniform on [0 1]. The domain H is defined by the equation r 1 + r 2 = y, together with the conditions 0 < r 1 < 1 and 0 < r 2 < 1. This defines a straight line segment, which intersects the r 1 and r 2 axis at y, with 0 < y < 2. Equation 3 now reads Q(y) = ds. which is the length of the line segment times some constant we don t worry about here. The length of the line segment plotted as a function of y is just a triangle with peak at y = 1, the triangular distribution. 4 H
5 (a) Integral over H (b) Resulting triangular distribution Figure 2: The sum of two uniform random numbers obeys a triangular distribution Convolution gives us an easy way to generate the Erlang distribution as it is defined as the distribution of a sum of exponentially distributed variables. Note that if K is large, we will always generate an approximately normal distribution, as the convolution is just the mean up to a multiplicative constant 5 Acceptance-rejection This is sometimes an easy and fast method to program. Suppose we want to generate uniform random numbers on [c 1]. We could generate r on [0 1], accept it if r c, reject it and try again otherwise. In pseudo-c: double f1(double c) { double r; while((double r = rand())<c); return r; } Let s compare this to our inverse transform technique: double f2(double c) { double r; r = c + (1-c)*rand(); return r; } Which one is faster? The chance that f1 will generate a wrong value precisely n times is given by p = (1 c)c n. The chance for getting n wrong values is c n and the chance of 5
6 getting the right value at the end is 1 c. The expected value for w, the number of times a wrong value is generated is thus < w >= n(1 c)c n = c/(1 c). n=0 Suppose now the functions are called N times. On average we have to call rand() N(1+ < w >) = N/(1 c) times, and every time we have to do a compare. Method 2, (f2) on the other hand always needs to call rand() only once, but it has to do an addition, subtraction, and a multiplication every time. Which is faster depends clearly on c. Let s work it out. Let T R be the computation time for rand(), T A is the time to do an addition, subtraction, and a multiplication and let T C be the time to do the comparison. If we denote the time spent per call for the two algorithms by T 1 and T 2 we have and T 1 = 1 1 c (T R + T C ) T 2 = T R + T A. The acceptance-rejection algorithm is faster if T 1 < T 2 which we can rewrite as For example if c = 0.1 we get c 1 c T R + T c 1 c < T A. T C + 0.1T R < T A which is probably satisfied if we use the LCM algorithm for rand() as it requires about the same time as T A. This method is of course not only applicable to uniform distributions. Here s a more realistic example. IQ s are normally distributed with mean of 100 and standard deviation 15. This is of course an approximation, and in particular N(x, 100, 15) can generate negative IQ s. So if we want to generate a sample of IQ s we could try to use the inverse transform technique for a cutoff normal distribution. However it is much simpler and faster to use acceptance-rejection here and just try again if you should get a negative value. In fact, the chance of this happening is only about 1 in ten billion! 6
APPENDIX 2.1 LINE AND SURFACE INTEGRALS
2 APPENDIX 2. LINE AND URFACE INTEGRAL Consider a path connecting points (a) and (b) as shown in Fig. A.2.. Assume that a vector field A(r) exists in the space in which the path is situated. Then the line
More informationStatistics 100A Homework 5 Solutions
Chapter 5 Statistics 1A Homework 5 Solutions Ryan Rosario 1. Let X be a random variable with probability density function a What is the value of c? fx { c1 x 1 < x < 1 otherwise We know that for fx to
More informationMain topics for the First Midterm Exam
Main topics for the First Midterm Exam The final will cover Sections.-.0, 2.-2.5, and 4.. This is roughly the material from first three homeworks and three quizzes, in addition to the lecture on Monday,
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 informationModel-building and parameter estimation
Luleå University of Technology Johan Carlson Last revision: July 27, 2009 Measurement Technology and Uncertainty Analysis - E7021E MATLAB homework assignment Model-building and parameter estimation Introduction
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its y-intercept. In Euclidean geometry (where
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 informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationGetting Started with Communications Engineering
1 Linear algebra is the algebra of linear equations: the term linear being used in the same sense as in linear functions, such as: which is the equation of a straight line. y ax c (0.1) Of course, if we
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 informationHaus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, ISBN:
MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 989. ISBN: 978032490207. Please use the following citation
More informationWelcome to Math 104. D. DeTurck. January 16, University of Pennsylvania. D. DeTurck Math A: Welcome 1 / 44
Welcome to Math 104 D. DeTurck University of Pennsylvania January 16, 2018 D. DeTurck Math 104 002 2018A: Welcome 1 / 44 Welcome to the course Math 104 Calculus I Topics: Quick review of Math 103 topics,
More informationSubstitutions and by Parts, Area Between Curves. Goals: The Method of Substitution Areas Integration by Parts
Week #7: Substitutions and by Parts, Area Between Curves Goals: The Method of Substitution Areas Integration by Parts 1 Week 7 The Indefinite Integral The Fundamental Theorem of Calculus, b a f(x) dx =
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3. The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by
More informationSolution to Assignment 3
The Chinese University of Hong Kong ENGG3D: Probability and Statistics for Engineers 5-6 Term Solution to Assignment 3 Hongyang Li, Francis Due: 3:pm, March Release Date: March 8, 6 Dear students, The
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 informationSequences and infinite series
Sequences and infinite series D. DeTurck University of Pennsylvania March 29, 208 D. DeTurck Math 04 002 208A: Sequence and series / 54 Sequences The lists of numbers you generate using a numerical method
More information1 Functions of many variables.
MA213 Sathaye Notes on Multivariate Functions. 1 Functions of many variables. 1.1 Plotting. We consider functions like z = f(x, y). Unlike functions of one variable, the graph of such a function has to
More information2 Discrete Dynamical Systems (DDS)
2 Discrete Dynamical Systems (DDS) 2.1 Basics A Discrete Dynamical System (DDS) models the change (or dynamics) of single or multiple populations or quantities in which the change occurs deterministically
More informationWed Feb The vector spaces 2, 3, n. Announcements: Warm-up Exercise:
Wed Feb 2 4-42 The vector spaces 2, 3, n Announcements: Warm-up Exercise: 4-42 The vector space m and its subspaces; concepts related to "linear combinations of vectors" Geometric interpretation of vectors
More informationWill Landau. Feb 21, 2013
Iowa State University Feb 21, 2013 Iowa State University Feb 21, 2013 1 / 31 Outline Iowa State University Feb 21, 2013 2 / 31 random variables Two types of random variables: Discrete random variable:
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationSolution to Proof Questions from September 1st
Solution to Proof Questions from September 1st Olena Bormashenko September 4, 2011 What is a proof? A proof is an airtight logical argument that proves a certain statement in general. In a sense, it s
More informationWhen they compared their results, they had an interesting discussion:
27 2.5 Making My Point A Solidify Understanding Task Zac and Sione were working on predicting the number of quilt blocks in this pattern: CC BY Camille King https://flic.kr/p/hrfp When they compared their
More informationMAS1302 Computational Probability and Statistics
MAS1302 Computational Probability and Statistics April 23, 2008 3. Simulating continuous random behaviour 3.1 The Continuous Uniform U(0,1) Distribution We have already used this random variable a great
More informationChapter 1. Root Finding Methods. 1.1 Bisection method
Chapter 1 Root Finding Methods We begin by considering numerical solutions to the problem f(x) = 0 (1.1) Although the problem above is simple to state it is not always easy to solve analytically. This
More information7.1 Indefinite Integrals Calculus
7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential
More informationHypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =
Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so] - Exactly what it says. We develop a hypothesis,
More informationChapter 4 Notes, Calculus I with Precalculus 3e Larson/Edwards
4.1 The Derivative Recall: For the slope of a line we need two points (x 1,y 1 ) and (x 2,y 2 ). Then the slope is given by the formula: m = y x = y 2 y 1 x 2 x 1 On a curve we can find the slope of a
More informationDIFFERENTIATION AND INTEGRATION PART 1. Mr C s IB Standard Notes
DIFFERENTIATION AND INTEGRATION PART 1 Mr C s IB Standard Notes In this PDF you can find the following: 1. Notation 2. Keywords Make sure you read through everything and the try examples for yourself before
More informationExam Question 10: Differential Equations. June 19, Applied Mathematics: Lecture 6. Brendan Williamson. Introduction.
Exam Question 10: June 19, 2016 In this lecture we will study differential equations, which pertains to Q. 10 of the Higher Level paper. It s arguably more theoretical than other topics on the syllabus,
More informationStatistics 3657 : Moment Approximations
Statistics 3657 : Moment Approximations Preliminaries Suppose that we have a r.v. and that we wish to calculate the expectation of g) for some function g. Of course we could calculate it as Eg)) by the
More informationF X (x) = P [X x] = x f X (t)dt. 42 Lebesgue-a.e, to be exact 43 More specifically, if g = f Lebesgue-a.e., then g is also a pdf for X.
10.2 Properties of PDF and CDF for Continuous Random Variables 10.18. The pdf f X is determined only almost everywhere 42. That is, given a pdf f for a random variable X, if we construct a function g by
More informationPhysics 509: Error Propagation, and the Meaning of Error Bars. Scott Oser Lecture #10
Physics 509: Error Propagation, and the Meaning of Error Bars Scott Oser Lecture #10 1 What is an error bar? Someone hands you a plot like this. What do the error bars indicate? Answer: you can never be
More informationCommon ontinuous random variables
Common ontinuous random variables CE 311S Earlier, we saw a number of distribution families Binomial Negative binomial Hypergeometric Poisson These were useful because they represented common situations:
More informationA review of probability theory
1 A review of probability theory In this book we will study dynamical systems driven by noise. Noise is something that changes randomly with time, and quantities that do this are called stochastic processes.
More informationGradient. x y x h = x 2 + 2h x + h 2 GRADIENTS BY FORMULA. GRADIENT AT THE POINT (x, y)
GRADIENTS BY FORMULA GRADIENT AT THE POINT (x, y) Now let s see about getting a formula for the gradient, given that the formula for y is y x. Start at the point A (x, y), where y x. Increase the x coordinate
More informationLecture 7: Statistics and the Central Limit Theorem. Philip Moriarty,
Lecture 7: Statistics and the Central Limit Theorem Philip Moriarty, philip.moriarty@nottingham.ac.uk NB Notes based heavily on lecture slides prepared by DE Rourke for the F32SMS module, 2006 7.1 Recap
More informationIntegration Using Tables and Summary of Techniques
Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13 Introduction We wrap up integration techniques by discussing the following topics:
More informationPhysics 110. Electricity and Magnetism. Professor Dine. Spring, Handout: Vectors and Tensors: Everything You Need to Know
Physics 110. Electricity and Magnetism. Professor Dine Spring, 2008. Handout: Vectors and Tensors: Everything You Need to Know What makes E&M hard, more than anything else, is the problem that the electric
More informationOrdinary Differential Equations (ODEs)
c01.tex 8/10/2010 22: 55 Page 1 PART A Ordinary Differential Equations (ODEs) Chap. 1 First-Order ODEs Sec. 1.1 Basic Concepts. Modeling To get a good start into this chapter and this section, quickly
More informationLecture 6, September 1, 2017
Engineering Mathematics Fall 07 Lecture 6, September, 07 Escape Velocity Suppose we have a planet (or any large near to spherical heavenly body) of radius R and acceleration of gravity at the surface of
More informationMATH Green s Theorem Fall 2016
MATH 55 Green s Theorem Fall 16 Here is a statement of Green s Theorem. It involves regions and their boundaries. In order have any hope of doing calculations, you must see the region as the set of points
More informationSometimes the domains X and Z will be the same, so this might be written:
II. MULTIVARIATE CALCULUS The first lecture covered functions where a single input goes in, and a single output comes out. Most economic applications aren t so simple. In most cases, a number of variables
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 4.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More information5.9 Representations of Functions as a Power Series
5.9 Representations of Functions as a Power Series Example 5.58. The following geometric series x n + x + x 2 + x 3 + x 4 +... will converge when < x
More informationChapter 4. Probability Distributions Continuous
1 Chapter 4 Probability Distributions Continuous Thus far, we have considered discrete pdfs (sometimes called probability mass functions) and have seen how that probability of X equaling a single number
More informationCHAPTER 1: Functions
CHAPTER 1: Functions 1.1: Functions 1.2: Graphs of Functions 1.3: Basic Graphs and Symmetry 1.4: Transformations 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus 1.6: Combining Functions
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 14: Continuous random variables Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/
More informationWarm-up Simple methods Linear recurrences. Solving recurrences. Misha Lavrov. ARML Practice 2/2/2014
Solving recurrences Misha Lavrov ARML Practice 2/2/2014 Warm-up / Review 1 Compute 100 k=2 ( 1 1 ) ( = 1 1 ) ( 1 1 ) ( 1 1 ). k 2 3 100 2 Compute 100 k=2 ( 1 1 ) k 2. Homework: find and solve problem Algebra
More informationRandom Number Generation. CS1538: Introduction to simulations
Random Number Generation CS1538: Introduction to simulations Random Numbers Stochastic simulations require random data True random data cannot come from an algorithm We must obtain it from some process
More informationPhysics 250 Green s functions for ordinary differential equations
Physics 25 Green s functions for ordinary differential equations Peter Young November 25, 27 Homogeneous Equations We have already discussed second order linear homogeneous differential equations, which
More informationGiven a sequence a 1, a 2,...of numbers, the finite sum a 1 + a 2 + +a n,wheren is an nonnegative integer, can be written
A Summations When an algorithm contains an iterative control construct such as a while or for loop, its running time can be expressed as the sum of the times spent on each execution of the body of the
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationC-1. Snezana Lawrence
C-1 Snezana Lawrence These materials have been written by Dr. Snezana Lawrence made possible by funding from Gatsby Technical Education projects (GTEP) as part of a Gatsby Teacher Fellowship ad-hoc bursary
More informationLINEAR MMSE ESTIMATION
LINEAR MMSE ESTIMATION TERM PAPER FOR EE 602 STATISTICAL SIGNAL PROCESSING By, DHEERAJ KUMAR VARUN KHAITAN 1 Introduction Linear MMSE estimators are chosen in practice because they are simpler than the
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 informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Quadratic Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR ANTHONY BROWN 31 Graphs of Quadratic Functions 3 Quadratic Equations In Chapter we looked at straight lines,
More informationREVIEW FOR EXAM III SIMILARITY AND DIAGONALIZATION
REVIEW FOR EXAM III The exam covers sections 4.4, the portions of 4. on systems of differential equations and on Markov chains, and..4. SIMILARITY AND DIAGONALIZATION. Two matrices A and B are similar
More informationLecture 11. Probability Theory: an Overveiw
Math 408 - Mathematical Statistics Lecture 11. Probability Theory: an Overveiw February 11, 2013 Konstantin Zuev (USC) Math 408, Lecture 11 February 11, 2013 1 / 24 The starting point in developing the
More informationFoundations of Math II Unit 5: Solving Equations
Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following
More informationMATH 312 Section 8.3: Non-homogeneous Systems
MATH 32 Section 8.3: Non-homogeneous Systems Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Outline Undetermined Coefficients 2 Variation of Parameter 3 Conclusions Undetermined Coefficients
More informationIn this lecture we calculate moments and products of inertia of some simple geometric figures. (Refer Slide Time: 0:22)
Engineering Mechanics Professor Manoj K Harbola Department of Physics Indian Institute of Technology Kanpur Module 4 Lecture No 37 Properties of plane surfaces VIII: second moment and product of an area,
More informationV. Graph Sketching and Max-Min Problems
V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More informationRandom Variate Generation
Random Variate Generation 28-1 Overview 1. Inverse transformation 2. Rejection 3. Composition 4. Convolution 5. Characterization 28-2 Random-Variate Generation General Techniques Only a few techniques
More informationLecture : The Indefinite Integral MTH 124
Up to this point we have investigated the definite integral of a function over an interval. In particular we have done the following. Approximated integrals using left and right Riemann sums. Defined the
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES We have: Seen how to interpret derivatives as slopes and rates of change Seen how to estimate derivatives of functions given by tables of values Learned how
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 information2 Analogies between addition and multiplication
Problem Analysis The problem Start out with 99% water. Some of the water evaporates, end up with 98% water. How much of the water evaporates? Guesses Solution: Guesses: Not %. 2%. 5%. Not 00%. 3%..0%..5%.
More informationSimulation. Where real stuff starts
1 Simulation Where real stuff starts ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3 What is a simulation?
More informationSTAT2201. Analysis of Engineering & Scientific Data. Unit 3
STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random
More informationECE 302 Division 2 Exam 2 Solutions, 11/4/2009.
NAME: ECE 32 Division 2 Exam 2 Solutions, /4/29. You will be required to show your student ID during the exam. This is a closed-book exam. A formula sheet is provided. No calculators are allowed. Total
More informationRegression, part II. I. What does it all mean? A) Notice that so far all we ve done is math.
Regression, part II I. What does it all mean? A) Notice that so far all we ve done is math. 1) One can calculate the Least Squares Regression Line for anything, regardless of any assumptions. 2) But, if
More informationEXAMPLES OF PROOFS BY INDUCTION
EXAMPLES OF PROOFS BY INDUCTION KEITH CONRAD 1. Introduction In this handout we illustrate proofs by induction from several areas of mathematics: linear algebra, polynomial algebra, and calculus. Becoming
More informationMATH 310, REVIEW SHEET 2
MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,
More information1 General Tips and Guidelines for Proofs
Math 20F, 2015SS1 / TA: Jor-el Briones / Sec: A01 / Handout Page 1 of 5 1 General Tips and Guidelines for Proofs Proofs, or really any question that asks you to justify or explain something, are a huge
More informationMath Boot Camp: Integration
Math Boot Camp: Integration You can skip this boot camp if you can answer the following questions: What is the line integral of 1 r 2 r along a radial path starting from r = and ending at r = R? Prove
More informationAPPM/MATH 4/5520 Solutions to Exam I Review Problems. f X 1,X 2. 2e x 1 x 2. = x 2
APPM/MATH 4/5520 Solutions to Exam I Review Problems. (a) f X (x ) f X,X 2 (x,x 2 )dx 2 x 2e x x 2 dx 2 2e 2x x was below x 2, but when marginalizing out x 2, we ran it over all values from 0 to and so
More information2.4 Log-Arithm-etic. A Practice Understanding Task
2.4 Log-Arithm-etic A Practice Understanding Task Abe and Mary are feeling good about their log rules and bragging about their mathematical prowess to all of their friends when this exchange occurs: CC
More information2 Continuous Random Variables and their Distributions
Name: Discussion-5 1 Introduction - Continuous random variables have a range in the form of Interval on the real number line. Union of non-overlapping intervals on real line. - We also know that for any
More informationFrom Random Numbers to Monte Carlo. Random Numbers, Random Walks, Diffusion, Monte Carlo integration, and all that
From Random Numbers to Monte Carlo Random Numbers, Random Walks, Diffusion, Monte Carlo integration, and all that Random Walk Through Life Random Walk Through Life If you flip the coin 5 times you will
More informationMITOCW ocw f99-lec05_300k
MITOCW ocw-18.06-f99-lec05_300k This is lecture five in linear algebra. And, it will complete this chapter of the book. So the last section of this chapter is two point seven that talks about permutations,
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin
More information=.55 = = 5.05
MAT1193 4c Definition of derivative With a better understanding of limits we return to idea of the instantaneous velocity or instantaneous rate of change. Remember that in the example of calculating the
More information6.1 Moment Generating and Characteristic Functions
Chapter 6 Limit Theorems The power statistics can mostly be seen when there is a large collection of data points and we are interested in understanding the macro state of the system, e.g., the average,
More information2 Two-Point Boundary Value Problems
2 Two-Point Boundary Value Problems Another fundamental equation, in addition to the heat eq. and the wave eq., is Poisson s equation: n j=1 2 u x 2 j The unknown is the function u = u(x 1, x 2,..., x
More informationIntroduction to Linear Algebra
Introduction to Linear Algebra Linear algebra is the algebra of vectors. In a course on linear algebra you will also learn about the machinery (matrices and reduction of matrices) for solving systems of
More informationSection 1.x: The Variety of Asymptotic Experiences
calculus sin frontera Section.x: The Variety of Asymptotic Experiences We talked in class about the function y = /x when x is large. Whether you do it with a table x-value y = /x 0 0. 00.0 000.00 or with
More informationLinear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4
Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix
More informationPower series and Taylor series
Power series and Taylor series D. DeTurck University of Pennsylvania March 29, 2018 D. DeTurck Math 104 002 2018A: Series 1 / 42 Series First... a review of what we have done so far: 1 We examined series
More informationand likewise fdy = and we have fdx = f((x, g(x))) 1 dx. (0.1)
On line integrals in R 2 and Green s formulae. How Cauchy s formulae follows by Green s. Suppose we have some curve in R 2 which can be parametrized by t (ζ 1 (t), ζ 2 (t)), where t is in some interval
More informationDerivatives and the Product Rule
Derivatives and the Product Rule James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline 1 Differentiability 2 Simple Derivatives
More information1.1 GRAPHS AND LINEAR FUNCTIONS
MATHEMATICS EXTENSION 4 UNIT MATHEMATICS TOPIC 1: GRAPHS 1.1 GRAPHS AND LINEAR FUNCTIONS FUNCTIONS The concept of a function is already familiar to you. Since this concept is fundamental to mathematics,
More information4.4 Uniform Convergence of Sequences of Functions and the Derivative
4.4 Uniform Convergence of Sequences of Functions and the Derivative Say we have a sequence f n (x) of functions defined on some interval, [a, b]. Let s say they converge in some sense to a function f
More informationData Analysis I. Dr Martin Hendry, Dept of Physics and Astronomy University of Glasgow, UK. 10 lectures, beginning October 2006
Astronomical p( y x, I) p( x, I) p ( x y, I) = p( y, I) Data Analysis I Dr Martin Hendry, Dept of Physics and Astronomy University of Glasgow, UK 10 lectures, beginning October 2006 4. Monte Carlo Methods
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 information