MA : Introductory Probability
|
|
- Jasper Norton
- 5 years ago
- Views:
Transcription
1 MA : Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky Spring 2017 David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
2 Cumulative Distribution Functions Definition Let X be a continuous real-valued random variable. Then the cumulative distribution function or c.d.f. of X is defined by the equation F(x) = P(X x) David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
3 Cumulative Distribution Functions Definition Let X be a continuous real-valued random variable. Then the cumulative distribution function or c.d.f. of X is defined by the equation F(x) = P(X x) Here F(x) accumulates (or, more simply, cumulates) all of the probability less than or equal to x. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
4 Cumulative Distribution Functions The density function and the cumulative distribution function are related in the following way. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
5 Cumulative Distribution Functions The density function and the cumulative distribution function are related in the following way. Theorem Let X be a continuous real-valued random variable with density function f (x). Then the function defined by F(x) = x f (t)dt, < x < is the cumulative distribution function of X. Furthermore, by the FTC, we have F (x) = f (x). David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
6 Example A real number is chosen at random from [0, 1] with uniform probability, and then this number is squared. Let X represent the result. 1 What is the c.d.f. of X? 2 What is the density function of X? David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
7 Example A real number is chosen at random from [0, 1] with uniform probability, and then this number is squared. Let X represent the result. 1 What is the c.d.f. of X? 2 What is the density function of X? Solution 0 if, x 0 F(x) = x if, 0 x 1 1 if, x 1. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
8 Example A real number is chosen at random from [0, 1] with uniform probability, and then this number is squared. Let X represent the result. 1 What is the c.d.f. of X? 2 What is the density function of X? Solution 0 if, x 0 F(x) = x if, 0 x 1 1 if, x 1. 0 if, x 0 1 f (x) = 2 if, 0 x 1 x 0 if, x 1. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
9 Consider a random variable defined to be the sum of two random real numbers chosen uniformly from [0, 1]. Let the random variables X and Y denote the two chosen real numbers. Define Z = X + Y. 1 What is the c.d.f. of Z? 2 What is the density function of Z? David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
10 Consider a random variable defined to be the sum of two random real numbers chosen uniformly from [0, 1]. Let the random variables X and Y denote the two chosen real numbers. Define Z = X + Y. 1 What is the c.d.f. of Z? 2 What is the density function of Z? Solution 0 if, z 0 z 2 2 if, 0 z 1 F Z (z) = 1 (2 z)2 2 if, 1 z 2 1 if, z 2. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
11 Consider a random variable defined to be the sum of two random real numbers chosen uniformly from [0, 1]. Let the random variables X and Y denote the two chosen real numbers. Define Z = X + Y. 1 What is the c.d.f. of Z? 2 What is the density function of Z? Solution 0 if, z 0 z 2 2 if, 0 z 1 F Z (z) = 1 (2 z)2 2 if, 1 z 2 1 if, z 2. 0 if, z 0 z if, 0 z 1 f (z) = 2 z if, 1 z 2 0 if, z 2. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
12 Example (Homework 2 Problem 9) Take a stick and break it into two pieces, choosing the break point at random from a uniform distribution. What is the probability that the larger piece is no more than 2.8 times as long as the smaller piece? Solution David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
13 Example (Homework 2 Problem 9) Take a stick and break it into two pieces, choosing the break point at random from a uniform distribution. What is the probability that the larger piece is no more than 2.8 times as long as the smaller piece? Solution ( ) ( ) David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
14 Example (Homework 2 Problem 12) Suppose you choose a real number X from the interval [a,b] with the density function f (x) = Cx, where C is a constant. a) Find C. b) Find P(E), where E = [a, b] is a subinterval of [a,b] (as a function of a and b). c) Find P(X > 8). d) Find P(X < 13). e) Find P(X 2 21X ). Solution David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
MA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationA&S 320: Mathematical Modeling in Biology
A&S 320: Mathematical Modeling in Biology David Murrugarra Department of Mathematics, University of Kentucky http://www.ms.uky.edu/~dmu228/as320/ Spring 2016 David Murrugarra (University of Kentucky) A&S
More informationMA 137: Calculus I for the Life Sciences
MA 137: Calculus I for the Life Sciences David Murrugarra Department of Mathematics, University of Kentucky http://www.ms.uky.edu/~ma137/ Spring 2018 David Murrugarra (University of Kentucky) MA 137: Lecture
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationMA 138: Calculus II for the Life Sciences
MA 138: Calculus II for the Life Sciences David Murrugarra Department of Mathematics, University of Kentucky. Spring 2016 David Murrugarra (University of Kentucky) MA 138: Section 11.4.2 Spring 2016 1
More informationSTAT 430/510: Lecture 10
STAT 430/510: Lecture 10 James Piette June 9, 2010 Updates HW2 is due today! Pick up your HW1 s up in stat dept. There is a box located right when you enter that is labeled "Stat 430 HW1". It ll be out
More informationProperties of Continuous Probability Distributions The graph of a continuous probability distribution is a curve. Probability is represented by area
Properties of Continuous Probability Distributions The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve. The curve is called the probability
More informationBeautiful homework # 4 ENGR 323 CESSNA Page 1/5
Beautiful homework # 4 ENGR 33 CESSNA Page 1/5 Problem 3-14 An operator records the time to complete a mechanical assembly to the nearest second with the following results. seconds 30 31 3 33 34 35 36
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationMATH 104: INTRODUCTORY ANALYSIS SPRING 2008/09 PROBLEM SET 8 SOLUTIONS
MATH 04: INTRODUCTORY ANALYSIS SPRING 008/09 PROBLEM SET 8 SOLUTIONS. Let f : R R be continuous periodic with period, i.e. f(x + ) = f(x) for all x R. Prove the following: (a) f is bounded above below
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationThere are two basic kinds of random variables continuous and discrete.
Summary of Lectures 5 and 6 Random Variables The random variable is usually represented by an upper case letter, say X. A measured value of the random variable is denoted by the corresponding lower case
More informationMAT 271E Probability and Statistics
MAT 7E Probability and Statistics Spring 6 Instructor : Class Meets : Office Hours : Textbook : İlker Bayram EEB 3 ibayram@itu.edu.tr 3.3 6.3, Wednesday EEB 6.., Monday D. B. Bertsekas, J. N. Tsitsiklis,
More informationMAT 271E Probability and Statistics
MAT 71E Probability and Statistics Spring 013 Instructor : Class Meets : Office Hours : Textbook : Supp. Text : İlker Bayram EEB 1103 ibayram@itu.edu.tr 13.30 1.30, Wednesday EEB 5303 10.00 1.00, Wednesday
More informationScience One Integral Calculus. January 9, 2019
Science One Integral Calculus January 9, 2019 Recap: What have we learned so far? The definite integral is defined as a limit of Riemann sums Riemann sums can be constructed using any point in a subinterval
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 informationThe Geometric Random Walk: More Applications to Gambling
MATH 540 The Geometric Random Walk: More Applications to Gambling Dr. Neal, Spring 2008 We now shall study properties of a random walk process with only upward or downward steps that is stopped after the
More informationStochastic Simulation
Stochastic Simulation APPM 7400 Lesson 3: Testing Random Number Generators Part II: Uniformity September 5, 2018 Lesson 3: Testing Random Number GeneratorsPart II: Uniformity Stochastic Simulation September
More informationIC 102: Data Analysis and Interpretation
IC 102: Data Analysis and Interpretation Instructor: Guruprasad PJ Dept. Aerospace Engineering Indian Institute of Technology Bombay Powai, Mumbai 400076 Email: pjguru@aero.iitb.ac.in Phone no.: 2576 7142
More informationMath438 Actuarial Probability
Math438 Actuarial Probability Jinguo Lian Department of Math and Stats Jan. 22, 2016 Continuous Random Variables-Part I: Definition A random variable X is continuous if its set of possible values is an
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 informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0
ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x
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 informationf (1 0.5)/n Z =
Math 466/566 - Homework 4. We want to test a hypothesis involving a population proportion. The unknown population proportion is p. The null hypothesis is p = / and the alternative hypothesis is p > /.
More informationChapter 3. Chapter 3 sections
sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional Distributions 3.7 Multivariate Distributions
More informationFinal Solutions Fri, June 8
EE178: Probabilistic Systems Analysis, Spring 2018 Final Solutions Fri, June 8 1. Small problems (62 points) (a) (8 points) Let X 1, X 2,..., X n be independent random variables uniformly distributed on
More informationGoal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS
AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x.
More informationMA2501 Numerical Methods Spring 2015
Norwegian University of Science and Technology Department of Mathematics MA5 Numerical Methods Spring 5 Solutions to exercise set 9 Find approximate values of the following integrals using the adaptive
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 1: Continuous Random Variables Section 4.1 Continuous Random Variables Section 4.2 Probability Distributions & Probability Density
More informationPROBABILITY DENSITY FUNCTIONS
PROBABILITY DENSITY FUNCTIONS P.D.F. CALCULATIONS Question 1 (***) The lifetime of a certain brand of battery, in tens of hours, is modelled by the f x given by continuous random variable X with probability
More informationStochastic Simulation
Stochastic Simulation Idea: probabilities samples Get probabilities from samples: X count x 1 n 1. x k total. n k m X probability x 1. n 1 /m. x k n k /m If we could sample from a variable s (posterior)
More informationVector Space Examples Math 203 Spring 2014 Myers. Example: S = set of all doubly infinite sequences of real numbers = {{y k } : k Z, y k R}
Vector Space Examples Math 203 Spring 2014 Myers Example: S = set of all doubly infinite sequences of real numbers = {{y k } : k Z, y k R} Define {y k } + {z k } = {y k + z k } and c{y k } = {cy k }. Example:
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 informationIntroductory Probability
Introductory Probability Conditional Probability: Bayes Probabilities Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Computing Bayes Probabilities Conditional Probability and
More informationThe area under a curve. Today we (begin to) ask questions of the type: How much area sits under the graph of f(x) = x 2 over the interval [ 1, 2]?
The area under a curve. Today we (begin to) ask questions of the type: How much area sits under the graph of f(x) = x 2 over the interval [ 1, 2]? Before we work on How we will figure out Why velocity
More informationPhysicsAndMathsTutor.com
PhysicsAndMathsTutor.com June 2005 6. A continuous random variable X has probability density function f(x) where 3 k(4 x x ), 0 x 2, f( x) = 0, otherwise, where k is a positive integer. 1 (a) Show that
More informationSTAT/MA 416 Answers Homework 6 November 15, 2007 Solutions by Mark Daniel Ward PROBLEMS
STAT/MA 4 Answers Homework November 5, 27 Solutions by Mark Daniel Ward PROBLEMS Chapter Problems 2a. The mass p, corresponds to neither of the first two balls being white, so p, 8 7 4/39. The mass p,
More informationChapter 7. Functions of Random Variables
Chapter 7. Functions of Random Variables Sections 7.2 -- 7.4: Functions of Discrete Random Variables, Method of Distribution Functions and Method of Transformations in One Dimension Jiaping Wang Department
More informationDefinition: A random variable X is a real valued function that maps a sample space S into the space of real numbers R. X : S R
Random Variables Definition: A random variable X is a real valued function that maps a sample space S into the space of real numbers R. X : S R As such, a random variable summarizes the outcome of an experiment
More informationPARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.
PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 1 = 1 + 1 x 1 3 x + 1. The point is that
More informationMA 777: Topics in Mathematical Biology
MA 777: Topics in Mathematical Biology David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma777/ Spring 2018 David Murrugarra (University of Kentucky) Lecture
More information1. I had a computer generate the following 19 numbers between 0-1. Were these numbers randomly selected?
Activity #10: Continuous Distributions Uniform, Exponential, Normal) 1. I had a computer generate the following 19 numbers between 0-1. Were these numbers randomly selected? 0.12374454, 0.19609266, 0.44248450,
More informationSTAT 414: Introduction to Probability Theory
STAT 414: Introduction to Probability Theory Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical Exercises
More informationIndependent Events. Two events are independent if knowing that one occurs does not change the probability of the other occurring
Independent Events Two events are independent if knowing that one occurs does not change the probability of the other occurring Conditional probability is denoted P(A B), which is defined to be: P(A and
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 informationThe random variable 1
The random variable 1 Contents 1. Definition 2. Distribution and density function 3. Specific random variables 4. Functions of one random variable 5. Mean and variance 2 The random variable A random variable
More informationRandom variables. DS GA 1002 Probability and Statistics for Data Science.
Random variables DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Motivation Random variables model numerical quantities
More informationAnswers and expectations
Answers and expectations For a function f(x) and distribution P(x), the expectation of f with respect to P is The expectation is the average of f, when x is drawn from the probability distribution P E
More informationData Analysis-I. Interpolation. Soon-Hyung Yook. December 4, Soon-Hyung Yook Data Analysis-I December 4, / 1
Data Analysis-I Interpolation Soon-Hyung Yook December 4, 2015 Soon-Hyung Yook Data Analysis-I December 4, 2015 1 / 1 Table of Contents Soon-Hyung Yook Data Analysis-I December 4, 2015 2 / 1 Introduction
More informationMath Practice Final - solutions
Math 151 - Practice Final - solutions 2 1-2 -1 0 1 2 3 Problem 1 Indicate the following from looking at the graph of f(x) above. All answers are small integers, ±, or DNE for does not exist. a) lim x 1
More informationLecture 3. Discrete Random Variables
Math 408 - Mathematical Statistics Lecture 3. Discrete Random Variables January 23, 2013 Konstantin Zuev (USC) Math 408, Lecture 3 January 23, 2013 1 / 14 Agenda Random Variable: Motivation and Definition
More informationExample A. Define X = number of heads in ten tosses of a coin. What are the values that X may assume?
Stat 400, section.1-.2 Random Variables & Probability Distributions notes by Tim Pilachowski For a given situation, or experiment, observations are made and data is recorded. A sample space S must contain
More informationName: Algebra 1 Section 3 Homework Problem Set: Introduction to Functions
Name: Algebra 1 Section 3 Homework Problem Set: Introduction to Functions Remember: To receive full credit, you must show all of your work and circle/box your final answers. If you run out of room for
More informationDue Date: Thursday, March 22, 2018
The Notebook Project AP Calculus AB This project is designed to improve study skills and organizational skills for a successful career in mathematics. You are to turn a composition notebook into a Go To
More informationMath Spring Practice for the Second Exam.
Math 4 - Spring 27 - Practice for the Second Exam.. Let X be a random variable and let F X be the distribution function of X: t < t 2 t < 4 F X (t) : + t t < 2 2 2 2 t < 4 t. Find P(X ), P(X ), P(X 2),
More informationThe final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.
Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic
More informationLecture 2 : CS6205 Advanced Modeling and Simulation
Lecture 2 : CS6205 Advanced Modeling and Simulation Lee Hwee Kuan 21 Aug. 2013 For the purpose of learning stochastic simulations for the first time. We shall only consider probabilities on finite discrete
More informationReview of Probability. CS1538: Introduction to Simulations
Review of Probability CS1538: Introduction to Simulations Probability and Statistics in Simulation Why do we need probability and statistics in simulation? Needed to validate the simulation model Needed
More informationSTAT Section 2.1: Basic Inference. Basic Definitions
STAT 518 --- Section 2.1: Basic Inference Basic Definitions Population: The collection of all the individuals of interest. This collection may be or even. Sample: A collection of elements of the population.
More informationMath 151. Rumbos Spring Solutions to Review Problems for Exam 1
Math 5. Rumbos Spring 04 Solutions to Review Problems for Exam. There are 5 red chips and 3 blue chips in a bowl. The red chips are numbered,, 3, 4, 5 respectively, and the blue chips are numbered,, 3
More informationMath123 Lecture 1. Dr. Robert C. Busby. Lecturer: Office: Korman 266 Phone :
Lecturer: Math1 Lecture 1 Dr. Robert C. Busby Office: Korman 66 Phone : 15-895-1957 Email: rbusby@mcs.drexel.edu Course Web Site: http://www.mcs.drexel.edu/classes/calculus/math1_spring0/ (Links are case
More informationChapter 11 Sampling Distribution. Stat 115
Chapter 11 Sampling Distribution Stat 115 1 Definition 11.1 : Random Sample (finite population) Suppose we select n distinct elements from a population consisting of N elements, using a particular probability
More informationChapter 2 notes from powerpoints
Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real
More informationTopic 3 - Discrete distributions
Topic 3 - Discrete distributions Basics of discrete distributions Mean and variance of a discrete distribution Binomial distribution Poisson distribution and process 1 A random variable is a function which
More informationComputing logarithms and other special functions
Computing logarithms and other special functions Mike Giles University of Oxford Mathematical Institute Napier 400 NAIS Symposium April 2, 2014 Mike Giles (Oxford) Computing special functions April 2,
More informationH 2 : otherwise. that is simply the proportion of the sample points below level x. For any fixed point x the law of large numbers gives that
Lecture 28 28.1 Kolmogorov-Smirnov test. Suppose that we have an i.i.d. sample X 1,..., X n with some unknown distribution and we would like to test the hypothesis that is equal to a particular distribution
More informationTopic 4: Continuous random variables
Topic 4: Continuous random variables Course 003, 2018 Page 0 Continuous random variables Definition (Continuous random variable): An r.v. X has a continuous distribution if there exists a non-negative
More informationEE 345 MIDTERM 2 Fall 2018 (Time: 1 hour 15 minutes) Total of 100 points
Problem (8 points) Name EE 345 MIDTERM Fall 8 (Time: hour 5 minutes) Total of points How many ways can you select three cards form a group of seven nonidentical cards? n 7 7! 7! 765 75 = = = = = = 35 k
More informationConvergence Concepts of Random Variables and Functions
Convergence Concepts of Random Variables and Functions c 2002 2007, Professor Seppo Pynnonen, Department of Mathematics and Statistics, University of Vaasa Version: January 5, 2007 Convergence Modes Convergence
More informationDay 2 Notes: Riemann Sums In calculus, the result of f ( x)
AP Calculus Unit 6 Basic Integration & Applications Day 2 Notes: Riemann Sums In calculus, the result of f ( x) dx is a function that represents the anti-derivative of the function f(x). This is also sometimes
More informationCH5 CH6(Sections 1 through 5) Homework Problems
550.40 CH5 CH6(Sections 1 through 5) Homework Problems 1. Part of HW #6: CH 5 P1. Let X be a random variable with probability density function f(x) = c(1 x ) 1 < x < 1 (a) What is the value of c? (b) What
More informationA&S 320: Mathematical Modeling in Biology
A&S 320: Mathematical Modeling in Biology David Murrugarra Department of Mathematics, University of Kentucky http://www.ms.uky.edu/~dmu228/as320/ These slides were modified from Matthew Macauley s lecture
More informationProbability theory for Networks (Part 1) CS 249B: Science of Networks Week 02: Monday, 02/04/08 Daniel Bilar Wellesley College Spring 2008
Probability theory for Networks (Part 1) CS 249B: Science of Networks Week 02: Monday, 02/04/08 Daniel Bilar Wellesley College Spring 2008 1 Review We saw some basic metrics that helped us characterize
More informationQuadratic function and equations Quadratic function/equations, supply, demand, market equilibrium
Exercises 8 Quadratic function and equations Quadratic function/equations, supply, demand, market equilibrium Objectives - know and understand the relation between a quadratic function and a quadratic
More informationChapter 4 - Lecture 3 The Normal Distribution
Chapter 4 - Lecture 3 The October 28th, 2009 Chapter 4 - Lecture 3 The Standard Chapter 4 - Lecture 3 The Standard Normal distribution is a statistical unicorn It is the most important distribution in
More informationHomework for MATH 4603 (Advanced Calculus I) Fall Homework 13: Due on Tuesday 15 December. Homework 12: Due on Tuesday 8 December
Homework for MATH 4603 (Advanced Calculus I) Fall 2015 Homework 13: Due on Tuesday 15 December 49. Let D R, f : D R and S D. Let a S (acc S). Assume that f is differentiable at a. Let g := f S. Show that
More informationMath 3339 Homework 6 (Sections )
Math 3339 Homework 6 (Sections 5. 5.4) Name: Key PeopleSoft ID: Instructions: Homework will NOT be accepted through email or in person. Homework must be submitted through CourseWare BEFORE the deadline.
More informationUnit #6 Basic Integration and Applications Homework Packet
Unit #6 Basic Integration and Applications Homework Packet For problems, find the indefinite integrals below.. x 3 3. x 3x 3. x x 3x 4. 3 / x x 5. x 6. 3x x3 x 3 x w w 7. y 3 y dy 8. dw Daily Lessons and
More informationThus f is continuous at x 0. Matthew Straughn Math 402 Homework 6
Matthew Straughn Math 402 Homework 6 Homework 6 (p. 452) 14.3.3, 14.3.4, 14.3.5, 14.3.8 (p. 455) 14.4.3* (p. 458) 14.5.3 (p. 460) 14.6.1 (p. 472) 14.7.2* Lemma 1. If (f (n) ) converges uniformly to some
More informationEE 302 Division 1. Homework 6 Solutions.
EE 3 Division. Homework 6 Solutions. Problem. A random variable X has probability density { C f X () e λ,,, otherwise, where λ is a positive real number. Find (a) The constant C. Solution. Because of the
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 informationMonte Carlo Integration II & Sampling from PDFs
Monte Carlo Integration II & Sampling from PDFs CS295, Spring 2017 Shuang Zhao Computer Science Department University of California, Irvine CS295, Spring 2017 Shuang Zhao 1 Last Lecture Direct illumination
More informationContinuous distributions
CHAPTER 7 Continuous distributions 7.. Introduction A r.v. X is said to have a continuous distribution if there exists a nonnegative function f such that P(a X b) = ˆ b a f(x)dx for every a and b. distribution.)
More informationBasic concepts of probability theory
Basic concepts of probability theory Random variable discrete/continuous random variable Transform Z transform, Laplace transform Distribution Geometric, mixed-geometric, Binomial, Poisson, exponential,
More informationMath 651 Introduction to Numerical Analysis I Fall SOLUTIONS: Homework Set 1
ath 651 Introduction to Numerical Analysis I Fall 2010 SOLUTIONS: Homework Set 1 1. Consider the polynomial f(x) = x 2 x 2. (a) Find P 1 (x), P 2 (x) and P 3 (x) for f(x) about x 0 = 0. What is the relation
More informationIEOR 4703: Homework 2 Solutions
IEOR 4703: Homework 2 Solutions Exercises for which no programming is required Let U be uniformly distributed on the interval (0, 1); P (U x) = x, x (0, 1). We assume that your computer can sequentially
More informationHW7 Solutions. f(x) = 0 otherwise. 0 otherwise. The density function looks like this: = 20 if x [10, 90) if x [90, 100]
HW7 Solutions. 5 pts.) James Bond James Bond, my favorite hero, has again jumped off a plane. The plane is traveling from from base A to base B, distance km apart. Now suppose the plane takes off from
More informationArkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan
2.4 Random Variables Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan By definition, a random variable X is a function with domain the sample space and range a subset of the
More informationDetermine whether the formula determines y as a function of x. If not, explain. Is there a way to look at a graph and determine if it's a function?
1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More informationHomework 5 Solutions
126/DCP126 Probability, Fall 214 Instructor: Prof. Wen-Guey Tzeng Homework 5 Solutions 5-Jan-215 1. Let the joint probability mass function of discrete random variables X and Y be given by { c(x + y) ifx
More information23. A force in the negative direction of an x-axis is applied for 27ms to a 0.40kg ball initially moving at 14m/s in the positive direction of the
23. A force in the negative direction of an x-axis is applied for 27ms to a 0.40kg ball initially moving at 14m/s in the positive direction of the axis. The force varies in magnitude, and the impulse has
More informationTopic 4: Continuous random variables
Topic 4: Continuous random variables Course 3, 216 Page Continuous random variables Definition (Continuous random variable): An r.v. X has a continuous distribution if there exists a non-negative function
More informationGeneral Random Variables
1/65 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Probability A general random variable is discrete, continuous, or mixed. A discrete random variable
More informationElastic Potential Energy
Elastic Potential Energy If you pull on a spring and stretch it, then you do work. That is because you are applying a force over a displacement. Your pull is the force and the amount that you stretch the
More informationUnit 4 Day 4 & 5. Piecewise Functions
Unit 4 Day 4 & 5 Piecewise Functions Warm Up 1. Why does the inverse variation have a vertical asymptote? 2. Graph. Find the asymptotes. Write the domain and range using interval notation. a. b. f(x)=
More information120 CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE. Figure 1.30: The graph of g(x) =x 2/3.
120 CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE Figure 1.30: The graph of g(x) =x 2/3. We shall return to local extrema and monotonic functions, and look at them in more depth in Section 3.2. 1.5.1 Exercises
More information