Chapter 5. Random variables

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 5. Random variables"

Transcription

1 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 x to denote the various values that X can take discrete vs. continuous random variables a random variable is discrete if can only take on a countable number of distinct values, and continuous if it is characterized by an infinite range of values within some interval probability distribution function the function that assigns probabilities to events in which the random variable X takes on its possible values probability mass function the probability distribution function of a discrete random variable, which assigns a probability to each of the distinct values of the variable (we tabulate each value x along with the associated probability P (X = x)) 1

2 probability density function the probability distribution function of a continuous random variable, whose graph is a continuous curve that describes the likelihood that X takes on values that lie in various interval ranges cumulative distribution function the function that produces values of P (X x) for each possible value x of a (discrete or continuous) random variable X properties of a probability mass function Since the values P (X = x) of a probability mass function are probabilities, each must be a number between 0 and 1 The sum of all the values of a probability mass function must equal 1 2

3 expected value (E(X), or µ) for any discrete random variable X, the ideal (long-run) average value that X takes after observing infinitely many independent repetitions of X; computed from its probability mass function as the sum of the products of the values of X with their associated probabilities: E(X) = µ = x P (X = x) variance (V ar(x), or σ 2 ) for any discrete random variable X, the expected value of the squared deviations from µ of the values of X; computed from its probability mass function: V ar(x) = σ 2 = (x µ) 2 P (X = x) standard deviation (SD(X), or σ) for any discrete random variable X, the square root of its variance: SD(X) = σ = V ar(x) 3

4 Expectation and risk Uncertainty is viewed by consumers as risky; for instance, which of these three options would you go for: (1) a coin toss that determines which of two indistinguishable envelopes you are given, one of which contains $200 while the other requires you to pay a $100 penalty; (2) a coin toss that determines which of two indistinguishable envelopes you are given, one of which contains $100 while the other is empty; or (3) a single envelope which is known to contain $10? risk loving The risk loving consumer ignores risk and will seek the prospect with the highest possible reward, even if it threatens a negative expected gain (this person selects option #1 above) risk neutral The risk neutral consumer ignores risk and will accept any prospect that offers a positive expected gain (this person selects option #2 above) risk averse The risk averse consumer expects a reward for taking a risk (this person selects option #3 above) 4

5 Combining random variables and portfolio returns Investors build portfolios by distributing money over several investment options, but the return on each option can be viewed as a random variable (as its actual future return is unpredictable); assessing the return on the entire portfolio requires understanding the joint distribution of multiple random variables If X and Y are two random variables, and a and b are constants, then the variable ax + by, called a weighted combination of X and Y, has the following characteristics: its expected value is and its variance is E(aX + by ) = a E(X) + b E(Y ) V ar(ax+by ) = a 2 V ar(x)+2ab Cov(X, Y )+b 2 V ar(y ) 5

6 Thus, if a portfolio consists of investing a fraction w A of one s money in investment A (w A is also called the weight of investment A), and the remaining fraction w B in investment B, then the rate of return R p of the portfolio is directly related to the rates of return on the two investments, R A and R B : since R p = w A R A + w B R B, we have that the expected return on the portfolio is E(R p ) = w A E(R A ) + w B E(R B ), while the portfolio variance is V ar(r p ) = w 2 A V ar(r A )+2w A w B Cov(R A, R B )+w 2 B V ar(r B ) and the portfolio standard deviation is SD(R p ) = V ar(r p ) 6

7 Binomial random variables Bernoulli process series of independent and identical trials of an experiment which has only two outcomes, Success and Failure, and for which the probability p of Success (and therefore also the probability q = 1 p of Failure) is the same on each trial binomial random variable counts the number of Successes in a string of n trials of a Bernoulli process binomial probability mass function For x = 0, 1,..., n, we have ( ) n P (X = x) = p x q n x n! = x x!(n x)! px q n x binomial parameters if X is a binomial random variable, then E(X) = µ = np V ar(x) = σ 2 = npq SD(X) = σ = npq 7

8 Poisson random variables Poisson process the number of Successes of a series of independent and identical trials of an experiment take place during an interval of time or within a region of space so that the probability of Success is the same in all time intervals or spatial regions with equal duration or size Poisson random variable counts the number of Successes of a Poisson process in some time interval or spatial region Poisson probability mass function where µ measures the mean number of Successes of the Poisson process in the given time interval or spatial region, we have, for x = 0, 1,..., that P (X = x) = e µ µ x x! Poisson parameters if X is a Poisson random variable, then E(X) = µ V ar(x) = σ 2 = µ SD(X) = σ = µ 8

9 Hypergeometric random variables hypergeometric process a sample of n individuals is randomly selected without replacement from a population of size N containing exactly S Successes, in which n is a significant fraction of the size of N (so that distinct selections in the process are not independent of each other, and do not have the same probability of selecting a Success) hypergeometric random variable counts the number of Successes selected in a hypergeometric process hypergeometric probability mass function where a population of N individuals contain exactly S Sucesses, we have, for x = 0, 1,..., n, that ( S N S ) P (X = x) = x)( n x ( N n) 9

10 hypergeometric parameters if X is a hypergeometric random variable, then E(X) = µ = n S N V ar(x) = σ 2 = n S N SD(X) = σ = n S N ( 1 S ) N n N N 1 ( 1 S N ) N n N 1 10

University of California, Los Angeles Department of Statistics. Random variables

University of California, Los Angeles Department of Statistics. Random variables University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.

More information

Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

Random Variable: A function that assigns numerical values to all the outcomes in the sample space. STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

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

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 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

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

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

Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables

Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics Yuming Jiang 1 Some figures taken from the web. Contents Basic Concepts Random Variables Discrete Random Variables Continuous Random

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

Sample Questions for Mastery #5

Sample Questions for Mastery #5 Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could

More information

4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables 4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

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

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

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions 1. The following table contains a probability distribution for a random variable X. a. Find the expected value (mean) of X. x 1 2

More information

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. 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

More information

Normal Distribution as an Approximation to the Binomial Distribution

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

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

Discrete and Continuous Random Variables. Summer 2003

Discrete and Continuous Random Variables. Summer 2003 Discrete and Continuous Random Variables Summer 003 Random Variables A random variable is a rule that assigns a numerical value to each possible outcome of a probabilistic experiment. We denote a random

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

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

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

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

東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文

東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文 東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文 指 導 教 授 楊 朝 棟 博 士 以 網 路 功 能 虛 擬 化 實 作 網 路 即 時 流 量 監 控 服 務 研 究 生 楊 曜 佑 中 華 民 國 一 零 四 年 五 月 摘 要 與 的 概 念 一 同 發 展 的, 是 指 利 用 虛 擬 化 的 技 術, 將 現 有 的 網 路 硬 體 設 備, 利 用 軟 體 來 取 代 其

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

Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

More information

Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

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

You flip a fair coin four times, what is the probability that you obtain three heads.

You flip a fair coin four times, what is the probability that you obtain three heads. Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

More information

Chapter 2, part 2. Petter Mostad

Chapter 2, part 2. Petter Mostad Chapter 2, part 2 Petter Mostad mostad@chalmers.se Parametrical families of probability distributions How can we solve the problem of learning about the population distribution from the sample? Usual procedure:

More information

Normal distribution. ) 2 /2σ. 2π σ

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

More information

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1 IEOR 4106: Introduction to Operations Research: Stochastic Models SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability Models, by Sheldon

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

Random variables, probability distributions, binomial random variable

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

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

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

Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

More information

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day.

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day. Review Exam 2 This is a sample of problems that would be good practice for the exam. This is by no means a guarantee that the problems on the exam will look identical to those on the exam but it should

More information

DISCRETE RANDOM VARIABLES

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

More information

4: Probability. What is probability? Random variables (RVs)

4: Probability. What is probability? Random variables (RVs) 4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

More information

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

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

More information

The Binomial Probability Distribution

The Binomial Probability Distribution The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

More information

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...

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

More information

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

More information

We have discussed the notion of probabilistic dependence above and indicated that dependence is

We have discussed the notion of probabilistic dependence above and indicated that dependence is 1 CHAPTER 7 Online Supplement Covariance and Correlation for Measuring Dependence We have discussed the notion of probabilistic dependence above and indicated that dependence is defined in terms of conditional

More information

Section 5 Part 2. Probability Distributions for Discrete Random Variables

Section 5 Part 2. Probability Distributions for Discrete Random Variables Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability

More information

Practice problems for Homework 11 - Point Estimation

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:

More information

STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE

STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE TROY BUTLER 1. Random variables and distributions We are often presented with descriptions of problems involving some level of uncertainty about

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

Stats on the TI 83 and TI 84 Calculator

Stats on the TI 83 and TI 84 Calculator Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and

More information

Review the following from Chapter 5

Review the following from Chapter 5 Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

More information

Probability distributions

Probability distributions Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

More information

Probability Distributions

Probability Distributions Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.

More information

2 Binomial, Poisson, Normal Distribution

2 Binomial, Poisson, Normal Distribution 2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability

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

Chapter 5 Discrete Probability Distribution. Learning objectives

Chapter 5 Discrete Probability Distribution. Learning objectives Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able

More information

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 (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

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

6.2. Discrete Probability Distributions

6.2. Discrete Probability Distributions 6.2. Discrete Probability Distributions Discrete Uniform distribution (diskreetti tasajakauma) A random variable X follows the dicrete uniform distribution on the interval [a, a+1,..., b], if it may attain

More information

Lecture 8: Continuous random variables, expectation and variance

Lecture 8: Continuous random variables, expectation and variance Lecture 8: Continuous random variables, expectation and variance Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde

More information

Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values

More information

Some special discrete probability distributions

Some special discrete probability distributions University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that

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

Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem 1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / October

More information

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 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

More information

Toss a coin twice. Let Y denote the number of heads.

Toss a coin twice. Let Y denote the number of heads. ! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

More information

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

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

More information

Stats Review Chapters 5-6

Stats Review Chapters 5-6 Stats Review Chapters 5-6 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

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 5: Normal Probability Distributions - Solutions

Chapter 5: Normal Probability Distributions - Solutions Chapter 5: Normal Probability Distributions - Solutions Note: All areas and z-scores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that

More information

Binomial Distribution n = 20, p = 0.3

Binomial Distribution n = 20, p = 0.3 This document will describe how to use R to calculate probabilities associated with common distributions as well as to graph probability distributions. R has a number of built in functions for calculations

More information

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,

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 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

More information

Chapter 4. iclicker Question 4.4 Pre-lecture. Part 2. Binomial Distribution. J.C. Wang. iclicker Question 4.4 Pre-lecture

Chapter 4. iclicker Question 4.4 Pre-lecture. Part 2. Binomial Distribution. J.C. Wang. iclicker Question 4.4 Pre-lecture Chapter 4 Part 2. Binomial Distribution J.C. Wang iclicker Question 4.4 Pre-lecture iclicker Question 4.4 Pre-lecture Outline Computing Binomial Probabilities Properties of a Binomial Distribution Computing

More information

Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams

Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Review for Final Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Histogram Boxplots Chapter 3: Set

More information

Expectations. Expectations. (See also Hays, Appendix B; Harnett, ch. 3).

Expectations. Expectations. (See also Hays, Appendix B; Harnett, ch. 3). Expectations Expectations. (See also Hays, Appendix B; Harnett, ch. 3). A. The expected value of a random variable is the arithmetic mean of that variable, i.e. E() = µ. As Hays notes, the idea of the

More information

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1,

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1, THE BINOMIAL THEOREM Pascal s Triangle and the Binomial Expansion Consider the following binomial expansions: (p + q) 0 1, (p+q) 1 p+q, (p + q) p +pq + q, (p + q) 3 p 3 +3p q+3pq + q 3, (p + q) 4 p 4 +4p

More information

4. Joint Distributions

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

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

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

Examination 110 Probability and Statistics Examination

Examination 110 Probability and Statistics Examination Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

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

Chapter 6 Random Variables

Chapter 6 Random Variables Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340-344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:

More information

The normal approximation to the binomial

The normal approximation to the binomial The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very

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

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

Chapter 4. Probability Distributions

Chapter 4. Probability Distributions Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive

More information

Exercises with solutions (1)

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

More information

IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem

IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have

More information

Probability Calculator

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

More information

Lecture 2: Discrete Distributions, Normal Distributions. Chapter 1

Lecture 2: Discrete Distributions, Normal Distributions. Chapter 1 Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables

More information

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

Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage 5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single

More information

1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset

1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset Department of Economics Financial Economics University of California, Berkeley Economics 136 November 9, 2003 Fall 2006 Economics 136: Financial Economics Section Notes for Week 11 1 Capital Allocation

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

Normal Approximation. Contents. 1 Normal Approximation. 1.1 Introduction. Anthony Tanbakuchi Department of Mathematics Pima Community College

Normal Approximation. Contents. 1 Normal Approximation. 1.1 Introduction. Anthony Tanbakuchi Department of Mathematics Pima Community College Introductory Statistics Lectures Normal Approimation To the binomial distribution Department of Mathematics Pima Community College Redistribution of this material is prohibited without written permission

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

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

Hypothesis Testing. Learning Objectives. After completing this module, the student will be able to

Hypothesis Testing. Learning Objectives. After completing this module, the student will be able to Hypothesis Testing Learning Objectives After completing this module, the student will be able to carry out a statistical test of significance calculate the acceptance and rejection region calculate and

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

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

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

More information