# 2WB05 Simulation Lecture 8: Generating random variables

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 2WB05 Simulation Lecture 8: Generating random variables Marko Boon January 7, 2013

2 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions

3 Generating random variables 3/36 How do we generate random variables? Sampling from continuous distributions Sampling from discrete distributions

4 Continuous distributions 4/36 Inverse Transform Method Let the random variable X have a continuous and increasing distribution function F. Denote the inverse of F by F 1. Then X can be generated as follows: Generate U from U(0, 1); Return X = F 1 (U). If F is not continuous or increasing, then we have to use the generalized inverse function F 1 (u) = min{x : F(x) u}.

5 Continuous distributions 5/36 Examples X = a + (b a)u is uniform on (a, b); X = ln(u)/λ is exponential with parameter λ; X = ( ln(u)) 1/a /λ is Weibull, parameters a and λ. Unfortunately, for many distribution functions we do not have an easy-to-use (closed-form) expression for the inverse of F.

6 Continuous distributions 6/36 Composition method This method applies when the distribution function F can be expressed as a mixture of other distribution functions F 1, F 2,..., F(x) = p i F i (x), where p i 0, i=1 p i = 1 The method is useful if it is easier to sample from the F i s than from F. First generate an index I such that P(I = i) = p i, i = 1, 2,... Generate a random variable X with distribution function F I. i=1

7 Continuous distributions 7/36 Examples Hyper-exponential distribution: F(x) = p 1 F 1 (x) + p 2 F 2 (x) + + p k F k (x), x 0, where F i (x) is the exponential distribution with parameter µ i, i = 1,..., k. Double-exponential (or Laplace) distribution: where f denotes the density of F. f (x) = 1 2 e x, x < 0; 1 2 e x, x 0,

8 Continuous distributions 8/36 Convolution method In some case X can be expressed as a sum of independent random variables Y 1,..., Y n, so X = Y 1 + Y Y n. where the Y i s can be generated more easily than X. Algorithm: Generate independent Y 1,..., Y n, each with distribution function G; Return X = Y Y n.

9 Continuous distributions 9/36 Example If X is Erlang distributed with parameters n and µ, then X can be expressed as a sum of n independent exponentials Y i, each with mean 1/µ. Algorithm: Generate n exponentials Y 1,..., Y n, each with mean µ; Set X = Y Y n. More efficient algorithm: Generate n uniform (0, 1) random variables U 1,..., U n ; Set X = ln(u 1 U 2 U n )/µ.

10 Continuous distributions 10/36 Acceptance-Rejection method Denote the density of X by f. This method requires a function g that majorizes f, g(x) f (x) for all x. Now g will not be a density, since c = g(x)dx 1. Assume that c <. Then h(x) = g(x)/c is a density. Algorithm: 1. Generate Y having density h; 2. Generate U from U(0, 1), independent of Y ; 3. If U f (Y )/g(y ), then set X = Y ; else go back to step 1. The random variable X generated by this algorithm has density f.

11 Continuous distributions 11/36 Validity of the Acceptance-Rejection method Note P(X x) = P(Y x Y accepted). Now, and thus, letting x gives P(Y x, Y accepted) = x f (y) g(y) h(y)dy = 1 x c f (y)dy, P(Y accepted) = 1 c. Hence, P(X x) = P(Y x, Y accepted) P(Y accepted) = x f (y)dy.

12 Continuous distributions Note that the number of iterations is geometrically distributed with mean c. 12/36 How to choose g? Try to choose g such that the random variable Y can be generated rapidly; The probability of rejection in step 3 should be small; so try to bring c close to 1, which mean that g should be close to f.

13 Continuous distributions 13/36 Example The Beta(4,3) distribution has density f (x) = 60x 3 (1 x) 2, 0 x 1. The maximal value of f occurs at x = 0.6, where f (0.6) = Thus, if we define then g majorizes f. Algorithm: 1. Generate Y and U from U(0, 1); g(x) = , 0 x 1, 2. If U 60Y 3 (1 Y ) 2, then set X = Y ; else reject Y and return to step

14 Continuous distributions 14/36 Normal distribution Methods: Acceptance-Rejection method Box-Muller method

15 Continuous distributions 15/36 Acceptance-Rejection method If X is N(0, 1), then the density of X is given by f (x) = 2 2π e x2 /2, x > 0. Now the function g(x) = 2e/πe x majorizes f. This leads to the following algorithm: 1. Generate an exponential Y with mean 1; 2. Generate U from U(0, 1), independent of Y ; 3. If U e (Y 1)2 /2, then accept Y ; else reject Y and return to step Return X = Y or X = Y, both with probability 1/2.

16 Continuous distributions 16/36 Box-Muller method If U 1 and U 2 are independent U(0, 1) random variables, then X 1 = 2 ln U 1 cos(2πu 2 ) X 2 = 2 ln U 1 sin(2πu 2 ) are independent standard normal random variables. This method is implemented in the function nextgaussian() in java.util.random

17 Discrete distributions 17/36 Discrete version of Inverse Transform Method Let X be a discrete random variable with probabilities P(X = x i ) = p i, i = 0, 1,..., i=0 p i = 1. To generate a realization of X, we first generate U from U(0, 1) and then set X = x i if i 1 j=0 p j U < i j=0 p j.

18 Discrete distributions So the algorithm is as follows: Generate U from U(0, 1); Determine the index I such that and return X = x I. I 1 j=0 p j U < I j=0 p j 18/36 The second step requires a search; for example, starting with I = 0 we keep adding 1 to I until we have found the (smallest) I such that I U < j=0 Note: The algorithm needs exactly one uniform random variable U to generate X; this is a nice feature if you use variance reduction techniques. p j

19 Discrete distributions 19/36 Array method: when X has a finite support Suppose p i = k i /100, i = 1,..., m, where k i s are integers with 0 k i 100 Construct array A[i], i = 1,..., 100 as follows: set A[i] = x 1 for i = 1,..., k 1 set A[i] = x 2 for i = k 1 + 1,..., k 1 + k 2, etc. Then, first sample a random index I from 1,..., 100: I = U and set X = A[I ]

20 Discrete distributions 20/36 Bernoulli Two possible outcomes of X (success or failure): P(X = 1) = 1 P(X = 0) = p. Algorithm: Generate U from U(0, 1); If U p, then X = 1; else X = 0.

21 Discrete distributions 21/36 Discrete uniform The possible outcomes of X are m, m + 1,..., n and they are all equally likely, so Algorithm: Generate U from U(0, 1); Set X = m + (n m + 1)U. P(X = i) = Note: No search is required, and compute (n m + 1) ahead. 1, i = m, m + 1,..., n. n m + 1

22 Discrete distributions 22/36 Geometric A random variable X has a geometric distribution with parameter p if P(X = i) = p(1 p) i, i = 0, 1, 2,... ; X is the number of failures till the first success in a sequence of Bernoulli trials with success probability p. Algorithm: Generate independent Bernoulli(p) random variables Y 1, Y 2,...; let I be the index of the first successful one, so Y I = 1; Set X = I 1. Alternative algorithm: Generate U from U(0, 1); Set X = ln(u)/ ln(1 p).

23 Discrete distributions 23/36 Binomial A random variable X has a binomial distribution with parameters n and p if ( ) n P(X = i) = p i (1 p) n i, i = 0, 1,..., n; i X is the number of successes in n independent Bernoulli trials, each with success probability p. Algorithm: Generate n Bernoulli( p) random variables Y 1,..., Y n ; Set X = Y 1 + Y Y n. Alternative algorithms can be derived by using the following results.

24 Discrete distributions Let Y 1, Y 2,... be independent geometric(p) random variables, and I the smallest index such that 24/36 I +1 (Y i + 1) > n. Then the index I has a binomial distribution with parameters n and p. i=1 Let Y 1, Y 2,... be independent exponential random variables with mean 1, and I the smallest index such that I +1 i=1 Y i n i + 1 > ln(1 p). Then the index I has a binomial distribution with parameters n and p.

25 Discrete distributions 25/36 Negative Binomial A random variable X has a negative-binomial distribution with parameters n and p if ( ) n + i 1 P(X = i) = p n (1 p) i, i = 0, 1, 2,... ; i X is the number of failures before the n-th success in a sequence of independent Bernoulli trials with success probability p. Algorithm: Generate n geometric( p) random variables Y 1,..., Y n ; Set X = Y 1 + Y Y n.

26 Discrete distributions 26/36 Poisson A random variable X has a Poisson distribution with parameter λ if P(X = i) = λi i! e λ, i = 0, 1, 2,... ; X is the number of events in a time interval of length 1 if the inter-event times are independent and exponentially distributed with parameter λ. Algorithm: Generate exponential inter-event times Y 1, Y 2,... with mean 1; let I be the smallest index such that Set X = I. I +1 i=1 Y i > λ;

27 Discrete distributions 27/36 Poisson (alternative) Generate U(0,1) random variables U 1, U 2,...; let I be the smallest index such that I +1 i=1 U i < e λ ; Set X = I.

28 Fitting distributions 28/36 Input of a simulation Specifying distributions of random variables (e.g., interarrival times, processing times) and assigning parameter values can be based on: Historical numerical data Expert opinion In practice, there is sometimes real data available, but often the only information of random variables that is available is their mean and standard deviation.

29 Fitting distributions Empirical data can be used to: construct empirical distribution functions and generate samples from them during the simulation; fit theoretical distributions and then generate samples from the fitted distributions. 29/36

30 Fitting distributions 30/36 Moment-fitting Obtain an approximating distribution by fitting a phase-type distribution on the mean, E(X), and the coefficient of variation, c X = σ X E(X), of a given positive random variable X, by using the following simple approach.

31 Moment-fitting 31/36 Coefficient of variation less than 1: Mixed Erlang If 0 < c X < 1, then fit an E k 1,k distribution as follows. If 1 k c2 X 1 k 1, for certain k = 2, 3,..., then the approximating distribution is with probability p (resp. 1 p) the sum of k 1 (resp. k) independent exponentials with common mean 1/µ. By choosing p = 1 ) (kc 2X 1 + c 2 k(1 + c 2 X ) k2 c 2 X, µ = k p E(X), X the E k 1,k distribution matches E(X) and c X.

32 Moment-fitting 32/36 Coefficient of variation greater than 1: Hyperexponential In case c X 1, fit a H 2 (p 1, p 2 ; µ 1, µ 2 ) distribution. Phase diagram for the H k (p 1,..., p k ; µ 1,..., µ k ) distribution: µ 1 p 1 1 p k k µ k

33 Moment-fitting But the H 2 distribution is not uniquely determined by its first two moments. In applications, the H 2 distribution with balanced means is often used. This means that the normalization p 1 µ 1 = p 2 µ 2 is used. The parameters of the H 2 distribution with balanced means and fitting E(X) and c X ( 1) are given by ( p 1 = 1 c 2 X ) 2 c 2 X + 1, p 2 = 1 p 1, µ 1 = 2p 1 E(X), µ 2 = 2p 2 E(X). 33/36

34 Moment-fitting In case c 2 X 0.5 one can also use a Coxian-2 distribution for a two-moment fit. 34/36 Phase diagram for the Coxian-k distribution: µ 1 µ 2 µ k 1 p 1 2 p 2 p k 1 k 1 p 1 1 p 2 1 p k 1 The following parameter set for the Coxian-2 is suggested: µ 1 = 2/E(X), p 1 = 0.5/c 2 X, µ 2 = µ 1 p 1.

35 Moment-fitting 35/36 Fitting nonnegative discrete distributions Let X be a random variable on the non-negative integers with mean E(X) and coefficient of variation c X. Then it is possible to fit a discrete distribution on E(X) and c X using the following families of distributions: Mixtures of Binomial distributions; Poisson distribution; Mixtures of Negative-Binomial distributions; Mixtures of geometric distributions. This fitting procedure is described in Adan, van Eenige and Resing (see Probability in the Engineering and Informational Sciences, 9, 1995, pp ).

36 Fitting distributions Adequacy of fit 36/36 Graphical comparison of fitted and empirical curves; Statistical tests (goodness-of-fit tests).

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

### Math 461 Fall 2006 Test 2 Solutions

Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two

### Notes on the Negative Binomial Distribution

Notes on the Negative Binomial Distribution John D. Cook October 28, 2009 Abstract These notes give several properties of the negative binomial distribution. 1. Parameterizations 2. The connection between

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

### Probability Generating Functions

page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence

### Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

### Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### Chapter 4 Expected Values

Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges

### Random Variate Generation (Part 3)

Random Variate Generation (Part 3) Dr.Çağatay ÜNDEĞER Öğretim Görevlisi Bilkent Üniversitesi Bilgisayar Mühendisliği Bölümü &... e-mail : cagatay@undeger.com cagatay@cs.bilkent.edu.tr Bilgisayar Mühendisliği

### Practice problems for Homework 11 - Point Estimation

Practice problems for Homework 11 - Point Estimation 1. (10 marks) Suppose we want to select a random sample of size 5 from the current CS 3341 students. Which of the following strategies is the best:

### INSURANCE RISK THEORY (Problems)

INSURANCE RISK THEORY (Problems) 1 Counting random variables 1. (Lack of memory property) Let X be a geometric distributed random variable with parameter p (, 1), (X Ge (p)). Show that for all n, m =,

### Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

### CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

### Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

### Lecture 7: Continuous Random Variables

Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

### 5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

### LECTURE 16. Readings: Section 5.1. Lecture outline. Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process

LECTURE 16 Readings: Section 5.1 Lecture outline Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process Number of successes Distribution of interarrival times The

### This document is published at www.agner.org/random, Feb. 2008, as part of a software package.

Sampling methods by Agner Fog This document is published at www.agner.org/random, Feb. 008, as part of a software package. Introduction A C++ class library of non-uniform random number generators is available

### Section 5.1 Continuous Random Variables: Introduction

Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,

### Tenth Problem Assignment

EECS 40 Due on April 6, 007 PROBLEM (8 points) Dave is taking a multiple-choice exam. You may assume that the number of questions is infinite. Simultaneously, but independently, his conscious and subconscious

### Generating Random Variables and Stochastic Processes

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Generating Random Variables and Stochastic Processes 1 Generating U(0,1) Random Variables The ability to generate U(0, 1) random variables

### Practice Problems #4

Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice

### Generating Random Variables and Stochastic Processes

Monte Carlo Simulation: IEOR E4703 c 2010 by Martin Haugh Generating Random Variables and Stochastic Processes In these lecture notes we describe the principal methods that are used to generate random

### 1.1 Introduction, and Review of Probability Theory... 3. 1.1.1 Random Variable, Range, Types of Random Variables... 3. 1.1.2 CDF, PDF, Quantiles...

MATH4427 Notebook 1 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 1 MATH4427 Notebook 1 3 1.1 Introduction, and Review of Probability

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### Chapter 10 Monte Carlo Methods

411 There is no result in nature without a cause; understand the cause and you will have no need for the experiment. Leonardo da Vinci (1452-1519) Chapter 10 Monte Carlo Methods In very broad terms one

### 1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

### e.g. arrival of a customer to a service station or breakdown of a component in some system.

Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be

### Homework 4 - KEY. Jeff Brenion. June 16, 2004. Note: Many problems can be solved in more than one way; we present only a single solution here.

Homework 4 - KEY Jeff Brenion June 16, 2004 Note: Many problems can be solved in more than one way; we present only a single solution here. 1 Problem 2-1 Since there can be anywhere from 0 to 4 aces, the

### 6. Jointly Distributed Random Variables

6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.

### Generating Random Variates

CHAPTER 8 Generating Random Variates 8.1 Introduction...2 8.2 General Approaches to Generating Random Variates...3 8.2.1 Inverse Transform...3 8.2.2 Composition...11 8.2.3 Convolution...16 8.2.4 Acceptance-Rejection...18

### 12.5: CHI-SQUARE GOODNESS OF FIT TESTS

125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability

### 4. Joint Distributions of Two Random Variables

4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### Exponential Distribution

Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1

### Continuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.

UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Neda Farzinnia, UCLA Statistics University of California,

### The Exponential Distribution

21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding

### Probability Calculator

Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that

### MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

### Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

### MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008

Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the

### ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

### Chapter 3 Joint Distributions

Chapter 3 Joint Distributions 3.6 Functions of Jointly Distributed Random Variables Discrete Random Variables: Let f(x, y) denote the joint pdf of random variables X and Y with A denoting the two-dimensional

### Queueing Systems. Ivo Adan and Jacques Resing

Queueing Systems Ivo Adan and Jacques Resing Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands March 26, 2015 Contents

### Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG820). December 15, 2012.

Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG8). December 15, 12. 1. (3p) The joint distribution of the discrete random variables X and

### 3 Multiple Discrete Random Variables

3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f

### FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

### Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

### Joint Distributions. Tieming Ji. Fall 2012

Joint Distributions Tieming Ji Fall 2012 1 / 33 X : univariate random variable. (X, Y ): bivariate random variable. In this chapter, we are going to study the distributions of bivariate random variables

### Sufficient Statistics and Exponential Family. 1 Statistics and Sufficient Statistics. Math 541: Statistical Theory II. Lecturer: Songfeng Zheng

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Sufficient Statistics and Exponential Family 1 Statistics and Sufficient Statistics Suppose we have a random sample X 1,, X n taken from a distribution

### Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

### Exploratory Data Analysis

Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

### General Sampling Methods

General Sampling Methods Reference: Glasserman, 2.2 and 2.3 Claudio Pacati academic year 2016 17 1 Inverse Transform Method Assume U U(0, 1) and let F be the cumulative distribution function of a distribution

### A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation

Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr CS 551, Fall 2015 CS 551, Fall 2015

### Common probability distributionsi Math 217/218 Probability and Statistics Prof. D. Joyce, 2016

Introduction. ommon probability distributionsi Math 7/8 Probability and Statistics Prof. D. Joyce, 06 I summarize here some of the more common distributions used in probability and statistics. Some are

### Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### Chapters 5. Multivariate Probability Distributions

Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are

### 3. Continuous Random Variables

3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

### Lecture 5 : The Poisson Distribution

Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

### Evaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand

Proceedings of the 2009 Industrial Engineering Research Conference Evaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand Yasin Unlu, Manuel D. Rossetti Department of

### Exercises with solutions (1)

Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal

### Normal Distribution as an Approximation to the Binomial Distribution

Chapter 1 Student Lecture Notes 1-1 Normal Distribution as an Approximation to the Binomial Distribution : Goals ONE TWO THREE 2 Review Binomial Probability Distribution applies to a discrete random variable

### Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8

### Homework # 3 Solutions

Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

EXAM 3, FALL 003 Please note: On a one-time basis, the CAS is releasing annotated solutions to Fall 003 Examination 3 as a study aid to candidates. It is anticipated that for future sittings, only the

### P(X = x k ) = 1 = k=1

74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems 5.1.1 Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k

### 1.5 / 1 -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de. 1.5 Transforms

.5 / -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de.5 Transforms Using different summation and integral transformations pmf, pdf and cdf/ccdf can be transformed in such a way, that even

### Network Traffic Distribution Derek McAvoy Wireless Technology Strategy Architect. March 5, 2010

Network Traffic Distribution Derek McAvoy Wireless Technology Strategy Architect March 5, 21 Data Growth is Exponential 2.5 x 18 98% 2 95% Traffic 1.5 1 9% 75% 5%.5 Data Traffic Feb 29 25% 1% 5% 2% 5 1

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### ST 371 (VIII): Theory of Joint Distributions

ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or

### Internet Dial-Up Traffic Modelling

NTS 5, Fifteenth Nordic Teletraffic Seminar Lund, Sweden, August 4, Internet Dial-Up Traffic Modelling Villy B. Iversen Arne J. Glenstrup Jens Rasmussen Abstract This paper deals with analysis and modelling

### Principle of Data Reduction

Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then

### 2. Discrete random variables

2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be

### PROBABILITY AND SAMPLING DISTRIBUTIONS

PROBABILITY AND SAMPLING DISTRIBUTIONS SEEMA JAGGI AND P.K. BATRA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 0 0 seema@iasri.res.in. Introduction The concept of probability

### Practice Problems for Homework #6. Normal distribution and Central Limit Theorem.

Practice Problems for Homework #6. Normal distribution and Central Limit Theorem. 1. Read Section 3.4.6 about the Normal distribution and Section 4.7 about the Central Limit Theorem. 2. Solve the practice

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

### DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced

### ECE302 Spring 2006 HW5 Solutions February 21, 2006 1

ECE3 Spring 6 HW5 Solutions February 1, 6 1 Solutions to HW5 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics

### Statistics 100A Homework 4 Solutions

Problem 1 For a discrete random variable X, Statistics 100A Homework 4 Solutions Ryan Rosario Note that all of the problems below as you to prove the statement. We are proving the properties of epectation

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### SPARE PARTS INVENTORY SYSTEMS UNDER AN INCREASING FAILURE RATE DEMAND INTERVAL DISTRIBUTION

SPARE PARS INVENORY SYSEMS UNDER AN INCREASING FAILURE RAE DEMAND INERVAL DISRIBUION Safa Saidane 1, M. Zied Babai 2, M. Salah Aguir 3, Ouajdi Korbaa 4 1 National School of Computer Sciences (unisia),

### Basics of Statistical Machine Learning

CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that