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

Size: px
Start display at page:

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

Transcription

1 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, f Y (y), y Λ Y. The conditional probability density that Y takes on the value y given that X is equal to x is denoted by f Y x (y) and is given by f Y x (y) = f(x, y) f X (x) (x, y) Λ X,Y, x Λ X and the conditional probability density that X takes on the value x given that Y is equal to y is denoted by f X y (x) and is given by f X y (x) = f(x, y) f Y (y) (x, y) Λ X,Y, y Λ Y Example: In a box, there are 4 white balls, 3 black balls, 2 red balls. We randomly select 2 balls without replacement. Define X = {number of white balls} Y = {number of black balls} (1) Find density function f(x, y) and marginal densities f X (x) and f Y (y). (2) Find the conditional probability P(Y 1 X = ). (3) Find the conditional probability P(X = 1 Y = 2).

2 HW MATH 461/561 Lecture Notes 15 2 Solution: (1) We can find P(X =, Y = ) = C2 2 C 9 2 = 1, P(X =, Y = 1) = C3 1C 2 1 C 9 2 = 6 P(X =, Y = 2) = C3 2 C 9 2 P(X = 1, Y = 1) = C4 1C 3 1 C 9 2 = 3, P(X = 1, Y = ) = C4 1C 2 1 C 9 2 = 12, P(X = 2, Y = ) = C4 2 C 9 2 Dist. and marginal dists table Y \X 1 2 f Y (y) = 8 = 6 (2) Thus, f X (x) 1 f Y (1) = f Y () = 2 6 f(, 1) f X () = 6/ 1/ = 3/5 f(, ) f X () = 1/ 1/ = 1/1 P(Y 1 X = ) = f Y () + f Y (1) = 3/5 + 1/1 = 7/1. P(X = 1 Y = 2) = f X 2 (1) = f(1, 2) f Y (2) = 3/ =

3 HW MATH 461/561 Lecture Notes 15 3 Example: Given the joint pdf Find (a) P(Y < 1 X = 1/2) (b) P(Y < 1 X < 1) Solution: (a) Since Thus, (b) Since f(x, y) = 2e (x+y) < x < y, f X (x) = x 2e x e y dy = 2e 2x x > f Y 1(y) = f(1 2, y) 2 f X ( 1 2 ) = 2e (y+1 2 ) = e y+1 2, 2e 2(1 2 ) P(Y < 1 X = 1/2) = P(X < 1) = < y e 1 2 y dy = ee y = 1 1 e 2e 2x dx = 1 e 2 = =.865 P(X < 1, Y < 1) = dy[ y 2e (x+y) dx] = dy{2e y [ e x y ]} = {2e y 2e 2y }dy = 1 2/e + e 2 = P(Y < 1 X < 1) = = Intuitively, a mean or an expected value of a rv X is X s central tendency or average. Definition:

4 HW MATH 461/561 Lecture Notes 15 4 Given a discrete random variable X with range space Λ = {x 1,, x n } and the distribution table. The distribution table of r.v X X x 1 x 2 x n p(x) p(x 1 ) p(x 2 ) p(x n ) The mean or expected value of X is defined by µ = E(X) = n x i p(x i ) Example: In a class A, there are total 2 students. The students scores of the final exam are divided into five categories. Let X be a rv which associates each student s score of the final exam with a specified category. X has following distribution table: i=1 X p(x) 3/2 6/2 3/2 3/2 5/2 Find the average category of the students scores of the final exam. Solution: E(X) = 1 3/ / / / /2 = 3.5 Example: A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let X equal the number of women chosen to fill the two positions.

5 HW MATH 461/561 Lecture Notes 15 5 (a) Find the probability distribution of X; (b) Find the average number of women who will be hired. Solution: (a). The sample space Ω = {ω 1 = {M 1 M 2 }, ω 2 = {M 1 M 3 }, ω 3 = {M 2 M 3 }, ω 4 = {W 1 M 1 }, ω 5 = {W 1 M 2 }, ω 6 = {W 1 M 3 }, ω 7 = {W 2 M 1 }, ω 8 = {W 2 M 2 }, ω 9 = {W 2 M 3 }, ω 1 = {W 1 W 2 }} and the range space Λ := {x 1 =, x 2 = 1, x 3 = 2}. p(x 1 ) = p() = P(X = ) = P({ω 1, ω 2, ω 3 }) = P({{M 1 M 2 }, {M 1 M 3 }, {M 2 M 3 }}) = 3 1, p(x 2 ) = p(1) = P(X = 1) = P({ω 4, ω 5, ω 6, ω 7, ω 8, ω 9 }) = P({{W 1 M 1 }, {W 1 M 2 }, {W 1 M 3 }, {W 2 M 1 }, {W 2 M 2 }, {W 2 M 3 }}) = 6 1, p(x 3 ) = p(2) = P(X = 2) = P({ω 1 = {W 1 W 2 }}) = 1 1.

6 HW MATH 461/561 Lecture Notes 15 6 Therefore, The distribution table of r.v X X 1 2 p(x) 3/1 6/1 1/1 (b) µ = E(X) = x i Λ x ip(x i ) = (3/1) + 1(6/1) + 2(1/1) = 8/1. Definition: Given a continuous random variable X with range space Λ and the density function: f(x), x Λ. The mean or expected value of X is defined by µ = E(X) = xf(x)dx Example: Let X be a rv which represents the lifetime of a bulb. X has pdf f(x) = λe λx, x >, where λ > is a constant. Find the average lifetime of a bulb. Solution: Using integration by part, we can get µ = E(X) = λxe λx dx Λ = 1/λ Theorem: Let X 1,, X n be rv s for which E(X i ) <, i = 1, 2,, n and a 1,, a n be constants. Then, E(a 1 X a n X n ) = a 1 E(X 1 ) + + a n E(X n ) Theorem: (1) Let X be a discrete rv with range space Λ = {x 1,, x n } and the distribution table.

7 HW MATH 461/561 Lecture Notes 15 7 The distribution table of r.v X X x 1 x 2 x n p(x) p(x 1 ) p(x 2 ) p(x n ) Let g be a function such that g(x i ) p(x i ) < x i Λ The mean of g(x) is given by Eg(X) = n g(x i )p(x i ) i=1 (2) Let Y be a continuous rv with range space Λ and the density function: h(y), y Λ. Let g be a function such that g(y) h(y)dy < y Λ The mean of g(y ) is given by Eg(Y ) = Λ g(y)h(y)dy

8 HW MATH 461/561 Lecture Notes 15 8 Theorem: If X and Y are two independent rv s and E(X) <,E(Y ) <, then, we have E(XY ) = E(X)E(Y ). Example: Let X, Y be two independent rv s with density functions : f X (x) = 2x, < x < 1 and f Y (y) = 1/2 < y < 2 Find E(2X 3Y ) 2 = E[(2X 3Y ) 2 ]. Solution: Since (2X 3Y ) 2 = 4X 2 12XY + 9Y 2, we have Since Thus, E(2X 3Y ) 2 = 4E(X 2 ) 12E(X)E(Y ) + 9E(Y 2 ). E(X) = E(Y ) = E(X 2 ) = E(Y 2 ) 2 2 x2xdx = 2/3 y(1/2)dy = 1 x 2 2xdx = 1/2 y 2 (1/2)dy = 4/3 E(2X 3Y ) 2 = 4(1/2) 12(2/3)1 + 9(4/3) = 6 Definition: Given a discrete random variable X with range space Λ = {x 1,, x n } and distribution table as follows:

9 HW MATH 461/561 Lecture Notes 15 9 The distribution table of r.v X X x 1 x 2 x n p(x) p(x 1 ) p(x 2 ) p(x n ) Let µ = E(X) = n i=1 x ip(x i ) be its mean. Then, the variance of X is denoted by σ 2 or Var(X), is defined by Var(X) = σ 2 = E(X µ) 2 = x i Λ(x i µ) 2 p(x i ) The standard deviation of X is defined as σ = σ 2. Remark: The variance of a rv X is a measurement to indicate the dispersion of the distribution of X. Example: A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let X equal the number of women chosen to fill the two positions. Find the mean and standard deviation of X. Solution: The sample space Ω = {ω 1 = {M 1 M 2 }, ω 2 = {M 1 M 3 }, ω 3 = {M 2 M 3 }, ω 4 = {W 1 M 1 }, ω 5 = {W 1 M 2 }, ω 6 = {W 1 M 3 }, ω 7 = {W 2 M 1 }, ω 8 = {W 2 M 2 }, ω 9 = {W 2 M 3 }, ω 1 = {W 1 W 2 }} and the range space Λ := {x 1 =, x 2 = 1, x 3 = 2}.

10 HW MATH 461/561 Lecture Notes 15 1 p(x 1 ) = p() = P(X = ) = P({ω 1, ω 2, ω 3 }) = P({{M 1 M 2 }, {M 1 M 3 }, {M 2 M 3 }}) = 3 1, p(x 2 ) = p(1) = P(X = 1) = P({ω 4, ω 5, ω 6, ω 7, ω 8, ω 9 }) = P({{W 1 M 1 }, {W 1 M 2 }, {W 1 M 3 }, {W 2 M 1 }, {W 2 M 2 }, {W 2 M 3 }}) = 6 1, p(x 3 ) = p(2) = P(X = 2) = P({ω 1 = {W 1 W 2 }}) = 1 1. Therefore, The distribution table of r.v X x 1 2 p(x) 3/1 6/1 1/1 The mean is µ = E(X) = x i Λ x ip(x i ) = (3/1) + 1(6/1) + 2(1/1) = 8/1. The variance of X is σ 2 = E[(X µ) 2 ] = x i Λ (x i µ) 2 p(x i ) = ( 8/1) 2 (3/1) + (1 8/1) 2 (6/1) + (2 8/1) 2 (1/1) =. and σ = σ 2 =.6

11 HW MATH 461/561 Lecture Notes Definition: Given a continuous random variable X with density function f(x), x Λ. Let µ = E(X) = Λ xf(x)dx be its mean. Then, the variance of X is denoted by σ 2 or Var(X), is defined by Var(X) = σ 2 = E(X µ) 2 = (x µ) 2 f(x)dx x Λ The standard deviation of X is defined as σ = σ 2. Theorem: Let X be a rv, discrete or continuous, have mean µ and finite E(X 2 ). Then, Var(X) = σ 2 = E(X µ) 2 = E(X 2 ) µ 2 Theorem: Let X be a rv, discrete or continuous, and a, b be two constants. Then, Var(aX + b) = a 2 Var(X) Theorem: Let X 1,, X n be independent rv s, discrete or continuous. Then, Var(X X n ) = Var(X 1 ) + + Var(X n ) Example: Let X,Y be two independent rv s with following pdf s: Find Var(7X + 8Y ). Solution: f X (x) = 2x, < x < 1 f Y (y) = 6y 5, < y < 1 E(X) = x2xdx = 2/3x 3 1 = 2/3

12 HW MATH 461/561 Lecture Notes E(Y ) = y6y 5 dy = 6/7x 7 1 = 6/7 Var(7X + 8Y ) = Var(7X) + Var(8Y ) = 7 2 Var(X) Var(Y ) Since Thus, = 49(EX 2 4/9) + 64(EY 2 /49) EX 2 = EY 2 = x 2 2xdx = 1/2 x 2 6x 5 dx = 3/4 Var(7X + 8Y ) = 49(EX 2 4/9) + 64(EY 2 /49) = 49(1/2 4/9) + 64(3/4 /49) = 3265/882 Theorem: Let X be a rv with binomial distribution b(n, p). Then, µ = E(X) = np and Var(X) = np(1 p). Example: Using Var(X X n ) = Var(X 1 ) + + Var(X n ) if X 1,, X n are independent, prove above theorem. Example: Let X 1,, X 1 be independent rv s and each with binomial distribution b(1, 1/3). Find and E{ i=1 X i } Var( i=1 X i)

Lecture 6: Discrete & Continuous Probability and Random Variables

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

More information

STAT 430/510 Probability Lecture 14: Joint Probability Distribution, Continuous Case

STAT 430/510 Probability Lecture 14: Joint Probability Distribution, Continuous Case STAT 430/510 Probability Lecture 14: Joint Probability Distribution, Continuous Case Pengyuan (Penelope) Wang June 20, 2011 Joint density function of continuous Random Variable When X and Y are two continuous

More information

Jointly Distributed Random Variables

Jointly Distributed Random Variables Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................

More information

Chapters 5. Multivariate Probability Distributions

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

More information

5. Continuous Random Variables

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

More information

Lecture Notes 1. Brief Review of Basic Probability

Lecture Notes 1. Brief Review of Basic Probability Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters -3 are a review. I will assume you have read and understood Chapters -3. Here is a very

More information

3 Multiple Discrete Random Variables

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

More information

4. Joint Distributions of Two Random Variables

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

More information

What is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference

What is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference 0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures

More information

6. Jointly Distributed Random Variables

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.

More information

Notes on Continuous Random Variables

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

More information

Section 5.1 Continuous Random Variables: Introduction

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,

More information

Random Variables. Chapter 2. Random Variables 1

Random Variables. Chapter 2. Random Variables 1 Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

More information

Joint Exam 1/P Sample Exam 1

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

More information

MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

More information

ST 371 (IV): Discrete Random Variables

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

More information

Math 461 Fall 2006 Test 2 Solutions

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

More information

Topic 4: Multivariate random variables. Multiple random variables

Topic 4: Multivariate random variables. Multiple random variables Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly

More information

Statistics 100A Homework 7 Solutions

Statistics 100A Homework 7 Solutions Chapter 6 Statistics A Homework 7 Solutions Ryan Rosario. A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 5 percent will purchase

More information

Joint Distributions. Tieming Ji. Fall 2012

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

More information

Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 5 Solutions

Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 5 Solutions Math 370/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 5 Solutions About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the

More information

ST 371 (VIII): Theory of Joint Distributions

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

More information

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

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

More information

RANDOM VARIABLES MATH CIRCLE (ADVANCED) 3/3/2013. 3 k) ( 52 3 )

RANDOM VARIABLES MATH CIRCLE (ADVANCED) 3/3/2013. 3 k) ( 52 3 ) RANDOM VARIABLES MATH CIRCLE (ADVANCED) //0 0) a) Suppose you flip a fair coin times. i) What is the probability you get 0 heads? ii) head? iii) heads? iv) heads? For = 0,,,, P ( Heads) = ( ) b) Suppose

More information

Math 431 An Introduction to Probability. Final Exam Solutions

Math 431 An Introduction to Probability. Final Exam Solutions Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

More information

MAS108 Probability I

MAS108 Probability I 1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

More information

For a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )

For a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i ) Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll

More information

Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions

Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

More information

An Introduction to Basic Statistics and Probability

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

More information

3. The Economics of Insurance

3. The Economics of Insurance 3. The Economics of Insurance Insurance is designed to protect against serious financial reversals that result from random evens intruding on the plan of individuals. Limitations on Insurance Protection

More information

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

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?

More information

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

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 ()

More information

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

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

More information

Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions)

Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions) Math 370, Actuarial Problemsolving Spring 008 A.J. Hildebrand Practice Test, 1/8/008 (with solutions) About this test. This is a practice test made up of a random collection of 0 problems from past Course

More information

Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

More information

Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22

Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22 Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability

More information

Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions

Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions Math 70, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

More information

Introduction to Probability

Introduction to Probability Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

More information

Roulette. Math 5 Crew. Department of Mathematics Dartmouth College. Roulette p.1/14

Roulette. Math 5 Crew. Department of Mathematics Dartmouth College. Roulette p.1/14 Roulette p.1/14 Roulette Math 5 Crew Department of Mathematics Dartmouth College Roulette p.2/14 Roulette: A Game of Chance To analyze Roulette, we make two hypotheses about Roulette s behavior. When we

More information

ECE302 Spring 2006 HW7 Solutions March 11, 2006 1

ECE302 Spring 2006 HW7 Solutions March 11, 2006 1 ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 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 where

More information

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ

More information

Example. A casino offers the following bets (the fairest bets in the casino!) 1 You get $0 (i.e., you can walk away)

Example. A casino offers the following bets (the fairest bets in the casino!) 1 You get $0 (i.e., you can walk away) : Three bets Math 45 Introduction to Probability Lecture 5 Kenneth Harris aharri@umich.edu Department of Mathematics University of Michigan February, 009. A casino offers the following bets (the fairest

More information

Experimental Uncertainty and Probability

Experimental Uncertainty and Probability 02/04/07 PHY310: Statistical Data Analysis 1 PHY310: Lecture 03 Experimental Uncertainty and Probability Road Map The meaning of experimental uncertainty The fundamental concepts of probability 02/04/07

More information

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

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

More information

Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304.  jones/courses/141 Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard

More information

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

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

More information

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1]. Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

More information

P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )

P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i ) Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =

More information

Lecture 7: Continuous Random Variables

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

More information

The problems of first TA class of Math144

The problems of first TA class of Math144 The problems of first TA class of Math144 T eaching Assistant : Liu Zhi Date : F riday, F eb 2, 27 Q1. According to the journal Chemical Engineering, an important property of a fiber is its water absorbency.

More information

Definition 6.1.1. A r.v. X has a normal distribution with mean µ and variance σ 2, where µ R, and σ > 0, if its density is f(x) = 1. 2σ 2.

Definition 6.1.1. A r.v. X has a normal distribution with mean µ and variance σ 2, where µ R, and σ > 0, if its density is f(x) = 1. 2σ 2. Chapter 6 Brownian Motion 6. Normal Distribution Definition 6... A r.v. X has a normal distribution with mean µ and variance σ, where µ R, and σ > 0, if its density is fx = πσ e x µ σ. The previous definition

More information

STAT 3502. x 0 < x < 1

STAT 3502. x 0 < x < 1 Solution - Assignment # STAT 350 Total mark=100 1. A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous

More information

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

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

More information

Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions

Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions Math 370, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

More information

Covariance and Correlation. Consider the joint probability distribution f XY (x, y).

Covariance and Correlation. Consider the joint probability distribution f XY (x, y). Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Section 5-2 Covariance and Correlation Consider the joint probability distribution f XY (x, y). Is there a relationship between X and Y? If so, what kind?

More information

STA 256: Statistics and Probability I

STA 256: Statistics and Probability I Al Nosedal. University of Toronto. Fall 2014 1 2 3 4 5 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and

More information

( ) = P Z > = P( Z > 1) = 1 Φ(1) = 1 0.8413 = 0.1587 P X > 17

( ) = P Z > = P( Z > 1) = 1 Φ(1) = 1 0.8413 = 0.1587 P X > 17 4.6 I company that manufactures and bottles of apple juice uses a machine that automatically fills 6 ounce bottles. There is some variation, however, in the amounts of liquid dispensed into the bottles

More information

MTH135/STA104: Probability

MTH135/STA104: Probability MTH135/STA14: Probability Homework # 8 Due: Tuesday, Nov 8, 5 Prof Robert Wolpert 1 Define a function f(x, y) on the plane R by { 1/x < y < x < 1 f(x, y) = other x, y a) Show that f(x, y) is a joint probability

More information

Chapter 4 Lecture Notes

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,

More information

Lecture 10: Depicting Sampling Distributions of a Sample Proportion

Lecture 10: Depicting Sampling Distributions of a Sample Proportion Lecture 10: Depicting Sampling Distributions of a Sample Proportion Chapter 5: Probability and Sampling Distributions 2/10/12 Lecture 10 1 Sample Proportion 1 is assigned to population members having a

More information

Chapter 4. Multivariate Distributions

Chapter 4. Multivariate Distributions 1 Chapter 4. Multivariate Distributions Joint p.m.f. (p.d.f.) Independent Random Variables Covariance and Correlation Coefficient Expectation and Covariance Matrix Multivariate (Normal) Distributions Matlab

More information

Measurements of central tendency express whether the numbers tend to be high or low. The most common of these are:

Measurements of central tendency express whether the numbers tend to be high or low. The most common of these are: A PRIMER IN PROBABILITY This handout is intended to refresh you on the elements of probability and statistics that are relevant for econometric analysis. In order to help you prioritize the information

More information

WHERE DOES THE 10% CONDITION COME FROM?

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

More information

2WB05 Simulation Lecture 8: Generating random variables

2WB05 Simulation Lecture 8: Generating random variables 2WB05 Simulation Lecture 8: Generating random variables Marko Boon http://www.win.tue.nl/courses/2wb05 January 7, 2013 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions Generating

More information

Notes 11 Autumn 2005

Notes 11 Autumn 2005 MAS 08 Probabilit I Notes Autumn 005 Two discrete random variables If X and Y are discrete random variables defined on the same sample space, then events such as X = and Y = are well defined. The joint

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

More information

3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.

3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved. 3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately

More information

6.041/6.431 Spring 2008 Quiz 2 Wednesday, April 16, 7:30-9:30 PM. SOLUTIONS

6.041/6.431 Spring 2008 Quiz 2 Wednesday, April 16, 7:30-9:30 PM. SOLUTIONS 6.4/6.43 Spring 28 Quiz 2 Wednesday, April 6, 7:3-9:3 PM. SOLUTIONS Name: Recitation Instructor: TA: 6.4/6.43: Question Part Score Out of 3 all 36 2 a 4 b 5 c 5 d 8 e 5 f 6 3 a 4 b 6 c 6 d 6 e 6 Total

More information

Martingale Ideas in Elementary Probability. Lecture course Higher Mathematics College, Independent University of Moscow Spring 1996

Martingale Ideas in Elementary Probability. Lecture course Higher Mathematics College, Independent University of Moscow Spring 1996 Martingale Ideas in Elementary Probability Lecture course Higher Mathematics College, Independent University of Moscow Spring 1996 William Faris University of Arizona Fulbright Lecturer, IUM, 1995 1996

More information

Lecture 5: Mathematical Expectation

Lecture 5: Mathematical Expectation Lecture 5: Mathematical Expectation Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 The Expected Value of a Random

More information

Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1

Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1 Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields

More information

Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value. Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

More information

Statistics 100A Homework 8 Solutions

Statistics 100A Homework 8 Solutions Part : Chapter 7 Statistics A Homework 8 Solutions Ryan Rosario. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, the one-half

More information

The Method of Lagrange Multipliers

The Method of Lagrange Multipliers The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,

More information

Particular Solutions. y = Ae 4x and y = 3 at x = 0 3 = Ae 4 0 3 = A y = 3e 4x

Particular Solutions. y = Ae 4x and y = 3 at x = 0 3 = Ae 4 0 3 = A y = 3e 4x Particular Solutions If the differential equation is actually modeling something (like the cost of milk as a function of time) it is likely that you will know a specific value (like the fact that milk

More information

ECE302 Spring 2006 HW3 Solutions February 2, 2006 1

ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 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

More information

Probability & Probability Distributions

Probability & Probability Distributions Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions

More information

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. 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............................................

More information

PSTAT 120B Probability and Statistics

PSTAT 120B Probability and Statistics - Week University of California, Santa Barbara April 10, 013 Discussion section for 10B Information about TA: Fang-I CHU Office: South Hall 5431 T Office hour: TBA email: chu@pstat.ucsb.edu Slides will

More information

Recursive Estimation

Recursive Estimation Recursive Estimation Raffaello D Andrea Spring 04 Problem Set : Bayes Theorem and Bayesian Tracking Last updated: March 8, 05 Notes: Notation: Unlessotherwisenoted,x, y,andz denoterandomvariables, f x

More information

Topic 8 The Expected Value

Topic 8 The Expected Value Topic 8 The Expected Value Functions of Random Variables 1 / 12 Outline Names for Eg(X ) Variance and Standard Deviation Independence Covariance and Correlation 2 / 12 Names for Eg(X ) If g(x) = x, then

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 18. A Brief Introduction to Continuous Probability

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 18. A Brief Introduction to Continuous Probability CS 7 Discrete Mathematics and Probability Theory Fall 29 Satish Rao, David Tse Note 8 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces

More information

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS Copyright 5 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

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

More information

STAT/MTHE 353: Probability II. STAT/MTHE 353: Multiple Random Variables. Review. Administrative details. Instructor: TamasLinder

STAT/MTHE 353: Probability II. STAT/MTHE 353: Multiple Random Variables. Review. Administrative details. Instructor: TamasLinder STAT/MTHE 353: Probability II STAT/MTHE 353: Multiple Random Variables Administrative details Instructor: TamasLinder Email: linder@mast.queensu.ca T. Linder ueen s University Winter 2012 O ce: Je ery

More information

Covariance and Correlation

Covariance and Correlation Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

More information

STAT 830 Convergence in Distribution

STAT 830 Convergence in Distribution STAT 830 Convergence in Distribution Richard Lockhart Simon Fraser University STAT 830 Fall 2011 Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 Fall 2011 1 / 31

More information

Sums of Independent Random Variables

Sums of Independent Random Variables Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables

More information

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

More information

An-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)

An-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211) An-Najah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 5 Example: Joint

More information

Statistics 104: Section 6!

Statistics 104: Section 6! Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

More information

The Normal Distribution. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University

The Normal Distribution. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University The Normal Distribution Alan T. Arnholt Department of Mathematical Sciences Appalachian State University arnholt@math.appstate.edu Spring 2006 R Notes 1 Copyright c 2006 Alan T. Arnholt 2 Continuous Random

More information

Mathematical Expectation

Mathematical Expectation Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

Statistics 100A Homework 3 Solutions

Statistics 100A Homework 3 Solutions Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win $ for each black ball selected and we

More information

Example 1: Dear Abby. Stat Camp for the Full-time MBA Program

Example 1: Dear Abby. Stat Camp for the Full-time MBA Program Stat Camp for the Full-time MBA Program Daniel Solow Lecture 4 The Normal Distribution and the Central Limit Theorem 188 Example 1: Dear Abby You wrote that a woman is pregnant for 266 days. Who said so?

More information

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

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,

More information

2. Discrete random variables

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

More information

Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions

Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the

More information

1. Let A, B and C are three events such that P(A) = 0.45, P(B) = 0.30, P(C) = 0.35,

1. Let A, B and C are three events such that P(A) = 0.45, P(B) = 0.30, P(C) = 0.35, 1. Let A, B and C are three events such that PA =.4, PB =.3, PC =.3, P A B =.6, P A C =.6, P B C =., P A B C =.7. a Compute P A B, P A C, P B C. b Compute P A B C. c Compute the probability that exactly

More information