Chapter 5. Random variables

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

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

Random Variable: A variable whose value is the numerical outcome of an experiment or random phenomenon.

Random Variable: A variable whose value is the numerical outcome of an experiment or random phenomenon. STAT 515 -- Chapter 4: Discrete Random Variables Random Variable: A variable whose value is the numerical outcome of an experiment or random phenomenon. Discrete Random Variable : A numerical r.v. that

More information

Section 5-2 Random Variables

Section 5-2 Random Variables Section 5-2 Random Variables 5.1-1 Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies

More information

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

MCQ BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS

MCQ BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS MCQ BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS MCQ 8.1 A Bernoulli trial has: (a) At least two outcomes (c) Two outcomes (b) At most two outcomes (d) Fewer than two outcomes MCQ 8.2 The two mutually exclusive

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

Discrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)

Discrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8) 3 Discrete Random Variables and Probability Distributions Stat 4570/5570 Based on Devore s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer

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

PROPERTIES OF PROBABILITY P (A B) = 0. P (A B) = P (A B)P (B).

PROPERTIES OF PROBABILITY P (A B) = 0. P (A B) = P (A B)P (B). PROPERTIES OF PROBABILITY S is the sample space A, B are arbitrary events, A is the complement of A Proposition: For any event A P (A ) = 1 P (A). Proposition: If A and B are mutually exclusive, that is,

More information

Discrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)

Discrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8) 3 Discrete Random Variables and Probability Distributions Stat 4570/5570 Based on Devore s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer

More information

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014 STAT511 Spring 2014 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution January 28, 2014 3 Discrete Random Variables Chapter Overview Random Variable (r.v. Definition Discrete

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

Chapter 6: Probability Distributions

Chapter 6: Probability Distributions Chapter 6: Probability Distributions Section 1: Random Variables and their Distributions Example: Toss a coin twice Outcome # of heads Probability HH 2 1/4 HT 1 1/4 TH 1 1/4 TT 0 1/4 Definition: A random

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

PROBABILITY DISTRIBUTIONS. Introduction to Probability Distributions

PROBABILITY DISTRIBUTIONS. Introduction to Probability Distributions PROBABILITY DISTRIBUTIONS Introduction to Probability Distributions What is a Probability Distribution? Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero

More information

1 Probability Distributions

1 Probability Distributions 1 Probability Distributions In the chapter about descriptive statistics samples were discussed, and tools introduced for describing the samples with numbers as well as with graphs. In this chapter models

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

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

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 00A Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.

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

Probability Space. Sample space (denoted by ) by the problem we concern. Event space (denoted by a -field)

Probability Space. Sample space (denoted by ) by the problem we concern. Event space (denoted by a -field) 4.1 Probability Space Sample space (denoted by ) by the problem we concern Event space (denoted by a -field) by a random variable preserving the structure Support (on the real line) with Copyright 013

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

Outline. Random variables. Discrete random variables. Discrete probability distributions. Continuous random variable. ACE 261 Fall 2002 Prof.

Outline. Random variables. Discrete random variables. Discrete probability distributions. Continuous random variable. ACE 261 Fall 2002 Prof. ACE 6 Fall 00 Prof. Katchova Lecture 5 Discrete Probability Distributions Outline Random variables Discrete probability distributions Epected value and variance Binomial probability distribution Poisson

More information

Lecture Notes: Variance, Law of Large Numbers, Central Limit Theorem

Lecture Notes: Variance, Law of Large Numbers, Central Limit Theorem Lecture Notes: Variance, Law of Large Numbers, Central Limit Theorem CS244-Randomness and Computation March 24, 2015 1 Variance Definition, Basic Examples The variance of a random variable is a measure

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

Chapter 6 The Binomial Probability Distribution and Related Topics

Chapter 6 The Binomial Probability Distribution and Related Topics Chapter 6 The Binomial Probability Distribution and Related Topics Statistical Experiments and Random Variables Statistical Experiments any process by which measurements are obtained. A quantitative variable,

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

Lecture 2. Discrete random variables

Lecture 2. Discrete random variables Lecture 2. Discrete random variables Mathematical Statistics and Discrete Mathematics November 4th, 2015 1 / 20 Random variables The setting that we start with is a sample space S and a probability measure

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

4. Random Variables. Many random processes produce numbers. These numbers are called random variables.

4. Random Variables. Many random processes produce numbers. These numbers are called random variables. 4. Random Variables Many random processes produce numbers. These numbers are called random variables. Examples (i) The sum of two dice. (ii) The length of time I have to wait at the bus stop for a #2 bus.

More information

Probability Distributions and Statistics

Probability Distributions and Statistics 8 Probability Distributions and Statistics Distributions of Random Variables Expected Value Variance and Standard Deviation Binomial Distribution Normal Distribution Applications of the Normal Distribution

More information

Random Variables and Their Expected Values

Random Variables and Their Expected Values Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution

More information

Probability Distribution

Probability Distribution Lecture 3 Probability Distribution Discrete Case Definition: A r.v. Y is said to be discrete if it assumes only a finite or countable number of distinct values. Definition: The probability that Y takes

More information

5.3: The Binomial Probability Distribution

5.3: The Binomial Probability Distribution 5.3: The Binomial Probability Distribution 5.3.1 Bernoulli trials: A Bernoulli trial is an experiment with exactly two possible outcomes. We refer to one of the outcomes as a success (S) and to the other

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

STATS8: Introduction to Biostatistics. Random Variables and Probability Distributions. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics. Random Variables and Probability Distributions. Babak Shahbaba Department of Statistics, UCI STATS8: Introduction to Biostatistics Random Variables and Probability Distributions Babak Shahbaba Department of Statistics, UCI Random variables In this lecture, we will discuss random variables and

More information

Chapter 3 - Lecture 6 Hypergeometric and Negative Binomial D. Distributions

Chapter 3 - Lecture 6 Hypergeometric and Negative Binomial D. Distributions Chapter 3 - Lecture 6 and Distributions October 14th, 2009 Chapter 3 - Lecture 6 and D experiment random variable distribution Moments Experiment Random Variable Distribution Moments and moment generating

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

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

Discrete Random Variables

Discrete Random Variables Discrete Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Random Variables 2 1.1 Introduction........................................ 2 1.2 Cumulative Distribution Function...........................

More information

5.0 Lesson Plan. Answer Questions. Expectation and Variance. Densities and Cumulative Distribution Functions. The Exponential Distribution

5.0 Lesson Plan. Answer Questions. Expectation and Variance. Densities and Cumulative Distribution Functions. The Exponential Distribution 5.0 Lesson Plan Answer Questions 1 Expectation and Variance Densities and Cumulative Distribution Functions The Exponential Distribution The Normal Approximation to the Binomial 5.1 Expectation and Variance

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

8.1 Distributions of Random Variables

8.1 Distributions of Random Variables 8.1 Distributions of Random Variables A random variable is a rule that assigns a number to each outcome of an experiment. We usually denote a random variable by X. There are 3 types of random variables:

More information

Properties of Point Estimators and Methods of Estimation

Properties of Point Estimators and Methods of Estimation Lecture 9 Properties of Point Estimators and Methods of Estimation Relative efficiency: If we have two unbiased estimators of a parameter, and, we say that is relatively more efficient than if ( ). Definition:

More information

Probability Concepts and Applications

Probability Concepts and Applications Chapter 2 Probability Concepts and Applications To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna Power Point slides created by Brian Peterson Learning Objectives

More information

Section Discrete and Continuous Random Variables. Random Variable and Probability Distribution

Section Discrete and Continuous Random Variables. Random Variable and Probability Distribution Section 7.2-7.3 Learning Objectives After this section, you should be able to APPLY the concept of discrete random variables to a variety of statistical settings CALCULATE and INTERPRET the mean (expected

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

Probability Distributions

Probability Distributions 9//5 : (Discrete) Random Variables For a given sample space S of some experiment, a random variable (RV) is any rule that associates a number with each outcome of S. In mathematical language, a random

More information

Learning Objectives. Sample: A sample is a subset of measurements selected from the population of interest. 1 P age

Learning Objectives. Sample: A sample is a subset of measurements selected from the population of interest. 1 P age Learning Objectives Definition: Statistics is a science, which deals with the collection of data, analysis of data, and making inferences about the population using the information contained in the sample.

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

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

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

5.1 Probability Distributions

5.1 Probability Distributions 5/3/03 Discrete Probability Distributions C H 5A P T E R Outline 5 Probability Distributions 5 Mean, Variance, Standard Deviation, and Expectation 5 3 The Binomial Distribution C H 5A P T E R Objectives

More information

Activity Duration Method Media References. 9. Evaluation The evaluation is performed based on the student activities in discussion, doing exercise.

Activity Duration Method Media References. 9. Evaluation The evaluation is performed based on the student activities in discussion, doing exercise. 5. Based Competency : To understand basic concepts of statistics a) Student can explain definition of statistics, data, types of data, variable b) Student can explain sample and population, parameter and

More information

Probability and Random Variables

Probability and Random Variables Probability and Random Variables Sanford Gordon February 28, 2003 1 The Sample Space Random experiment: repeatable procedure with uncertain outcome. Sample outcome: A possible observation of the experiment,

More information

Example 1. Consider tossing a coin n times. If X is the number of heads obtained, X is a random variable.

Example 1. Consider tossing a coin n times. If X is the number of heads obtained, X is a random variable. 7 Random variables A random variable is a real-valued function defined on some sample space. associates to each elementary outcome in the sample space a numerical value. That is, it Example 1. Consider

More information

The cumulative distribution function (c.d.f) (or simply the distribution function) of the random variable X, sayitf, is a function defined by

The cumulative distribution function (c.d.f) (or simply the distribution function) of the random variable X, sayitf, is a function defined by 2 Random Variables 2.1 Random variables Real valued-functions defined on the sample space, are known as random variables (r.v. s): RV : S R Example. X is a randomly selected number from a set of 1, 2,

More information

Data Modeling & Analysis Techniques. Probability Distributions. Manfred Huber

Data Modeling & Analysis Techniques. Probability Distributions. Manfred Huber Data Modeling & Analysis Techniques Probability Distributions Manfred Huber 2017 1 Experiment and Sample Space A (random) experiment is a procedure that has a number of possible outcomes and it is not

More information

Discrete Probability Distribution discrete continuous

Discrete Probability Distribution discrete continuous CHAPTER 5 Discrete Probability Distribution Objectives Construct a probability distribution for a random variable. Find the mean, variance, and expected value for a discrete random variable. Find the exact

More information

Chapter 4: Discrete Random Variables and the Binomial Distribution

Chapter 4: Discrete Random Variables and the Binomial Distribution Chapter 4: Discrete Random Variables and the Binomial Distribution Keith E. Emmert Department of Mathematics Tarleton State University June 16, 2011 Outline 1 Random Variables 2 Random Variables Some Basic

More information

2. Discrete Random Variables

2. Discrete Random Variables 2. Discrete Random Variables 2.1 Definition of a Random Variable A random variable is the numerical description of the outcome of an experiment (or observation). e.g. 1. The result of a die roll. 2. The

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

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

The Binomial Distribution

The Binomial Distribution The Binomial Distribution Binomial Experiment An experiment with these characteristics: 1 For some predetermined number n, a sequence of n smaller experiments called trials; 2 Each trial has two outcomes

More information

Introduction to Probability and Statistics Slides 3 Chapter 3

Introduction to Probability and Statistics Slides 3 Chapter 3 Introduction to Probability and Statistics Slides 3 Chapter 3 Ammar M. Sarhan, asarhan@mathstat.dal.ca Department of Mathematics and Statistics, Dalhousie University Fall Semester 2008 Dr. Ammar M. Sarhan

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

Practice Problems in Probability. 1 Easy and Medium Difficulty Problems

Practice Problems in Probability. 1 Easy and Medium Difficulty Problems Practice Problems in Probability 1 Easy and Medium Difficulty Problems Problem 1. Suppose we flip a fair coin once and observe either T for tails or H for heads. Let X 1 denote the random variable that

More information

Probability. Part 3 - Random Variables, Distributions and Expected Values. 1. References. (1) R. Durrett, The Essentials of Probability, Duxbury.

Probability. Part 3 - Random Variables, Distributions and Expected Values. 1. References. (1) R. Durrett, The Essentials of Probability, Duxbury. Probability Part 3 - Random Variables, Distributions and Expected Values 1. References (1) R. Durrett, The Essentials of Probability, Duxbury. (2) L.L. Helms, Probability Theory with Contemporary Applications,

More information

PROBABILITY DISTRIBUTIO S

PROBABILITY DISTRIBUTIO S CHAPTER 3 PROBABILITY DISTRIBUTIO S Page Contents 3.1 Introduction to Probability Distributions 51 3.2 The Normal Distribution 56 3.3 The Binomial Distribution 60 3.4 The Poisson Distribution 64 Exercise

More information

Discrete Random Variables

Discrete Random Variables Chapter 4 Discrete Random Variables 4.1 Discrete Random Variables 1 4.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand discrete probability

More information

Chapter 3. Probability Distributions (Part-II)

Chapter 3. Probability Distributions (Part-II) Chapter 3 Probability Distributions (Part-II) In Part-I of Chapter 3, we studied the normal and the t-distributions, which are the two most well-known distributions for continuous variables. When we are

More information

A continuous random variable can take on any value in a specified interval or range

A continuous random variable can take on any value in a specified interval or range Continuous Probability Distributions A continuous random variable can take on any value in a specified interval or range Example: Let X be a random variable indicating the blood level of serum triglycerides,

More information

7.1: Discrete and Continuous Random Variables

7.1: Discrete and Continuous Random Variables 7.1: Discrete and Continuous Random Variables RANDOM VARIABLE A random variable is a variable whose value is a numerical outcome of a random phenomenon. DISCRETE RANDOM VARIABLE A discrete random variable

More information

DATA ANALYSIS I. Data: Probabilistic View

DATA ANALYSIS I. Data: Probabilistic View DATA ANALYSIS I Data: Probabilistic View Sources Leskovec, J., Rajaraman, A., Ullman, J. D. (2014). Mining of massive datasets. Cambridge University Press. [5-7] Zaki, M. J., Meira Jr, W. (2014). Data

More information

Chapter 6 Random Variables and the Normal Distribution

Chapter 6 Random Variables and the Normal Distribution 1 Chapter 6 Random Variables and the Normal Distribution Random Variable o A random variable is a variable whose values are determined by chance. Discrete and Continuous Random Variables o A discrete random

More information

Fundamentals of Traffic Operations and Control Topic: Statistics for Traffic Engineers

Fundamentals of Traffic Operations and Control Topic: Statistics for Traffic Engineers Fundamentals of Traffic Operations and Control Topic: Statistics for Traffic Engineers Nikolas Geroliminis Ecole Polytechnique Fédérale de Lausanne nikolas.geroliminis@epfl.ch Role of Statistical Inference

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

Stat 104 Lecture 13. Statistics 104 Lecture 13 (IPS 4.3, 4.4 & 5.1) Random variables

Stat 104 Lecture 13. Statistics 104 Lecture 13 (IPS 4.3, 4.4 & 5.1) Random variables Statistics 104 Lecture 13 (IPS 4.3, 4.4 & 5.1) Random variables A random variable is a variable whose value is a numerical outcome of a random phenomenon Usually denoted by capital letters, Y or Z Example:

More information

Discrete Random Variables

Discrete Random Variables Discrete Random Variables A dichotomous random variable takes only the values 0 and 1. Let X be such a random variable, with Pr(X=1) = p and Pr(X=0) = 1-p. Then E[X] = p, and Var[X] = p(1-p). Consider

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

Continuous Random Variables (Devore Chapter Four)

Continuous Random Variables (Devore Chapter Four) Continuous Random Variables (Devore Chapter Four) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Continuous Random Variables 1 1.1 Defining the Probability Density Function...................

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

Discrete Probability Distributions. Chapter 6

Discrete Probability Distributions. Chapter 6 Discrete Probability Distributions Chapter 6 Learning Objectives Define terms random variable and probability distribution. Distinguish between discrete and continuous probability distributions. Calculate

More information

Probability Distributions for Discrete Random Variables. A discrete RV has a finite list of possible outcomes.

Probability Distributions for Discrete Random Variables. A discrete RV has a finite list of possible outcomes. 1 Discrete Probability Distributions Section 4.1 Random Variable- its value is a numerical outcome of a random phenomenon. We use letters such as x or y to denote a random variable. Examples Toss a coin

More information

RANDOM VARIABLES, EXPECTATION, AND VARIANCE

RANDOM VARIABLES, EXPECTATION, AND VARIANCE RANDOM VARIABLES, EXPECTATION, AND VARIANCE MATH 70 This write-up was originally created when we were using a different textbook in Math 70. It s optional, but the slightly different style may help you

More information

Random Variables. M. George Akritas. Outline Random Variables and Their Distribution The Expected Value of Discrete Random Variables

Random Variables. M. George Akritas. Outline Random Variables and Their Distribution The Expected Value of Discrete Random Variables Random Variables M. George Akritas Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function Definition Discrete and Continuous Random Variables The

More information

List of topics. Business Statistics Fact Sheet. basic probability. mutually exclusive versus independent. odds versus probability

List of topics. Business Statistics Fact Sheet. basic probability. mutually exclusive versus independent. odds versus probability Business Statistics 41000 Fact Sheet List of topics basic probability mutually exclusive versus independent odds versus probability law of total probability independence Bayes rule expected value variance/standard

More information

Properties of Expected values and Variance

Properties of Expected values and Variance Properties of Expected values and Variance Christopher Croke University of Pennsylvania Math 115 UPenn, Fall 2011 Expected value Consider a random variable Y = r(x ) for some function r, e.g. Y = X 2 +

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

Math 141. Lecture 5: Expected Value. Albyn Jones 1. jones/courses/ Library 304. Albyn Jones Math 141

Math 141. Lecture 5: Expected Value. Albyn Jones 1.  jones/courses/ Library 304. Albyn Jones Math 141 Math 141 Lecture 5: Expected Value Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 History The early history of probability theory is intimately related to questions arising

More information

Discrete random variables

Discrete random variables Chapter 4 Discrete random variables 4.1 Definition, Mean and Variance Let X be a (discrete) r.v. taking on the values x i with corresponding probabilities p i = f(x i ), i = 1,...,n. Function f(x) is called

More information

7.4 Expected Value and Variance

7.4 Expected Value and Variance 7.4 Expected Value and Variance Recall: A random variable is a function from the sample space of an experiment to the set of real numbers. That is, a random variable assigns a real number to each possible

More information

MAT S5.1 2_3 Random Variables. February 09, Review and Preview. Chapter 5 Probability Distributions. Preview

MAT S5.1 2_3 Random Variables. February 09, Review and Preview. Chapter 5 Probability Distributions. Preview MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance and Standard

More information

Random Variables. Definition Any random variable whose only possible values are 0 abd 1 is called a Bernoulli random variable.

Random Variables. Definition Any random variable whose only possible values are 0 abd 1 is called a Bernoulli random variable. Random Variables Definition For a given sample sample space S of some experiment, a random variable (rv) is any rule that associates a number with each outcome in S. In mathematical language, a random

More information

6.1. Discrete Random Variables

6.1. Discrete Random Variables 6.1 Discrete Random Variables Random Variables A random variable is a numeric measure of the outcome of a probability experiment Random variables reflect measurements that can change as the experiment

More information

Def: A random variable, x, represents a numerical value, determined by chance, assigned to an outcome of a probability experiment.

Def: A random variable, x, represents a numerical value, determined by chance, assigned to an outcome of a probability experiment. Lecture #5 chapter 5 Discrete Probability Distributions 5-2 Random Variables Def: A random variable, x, represents a numerical value, determined by chance, assigned to an outcome of a probability experiment.

More information

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution Chapter 6: Random Variables and the Normal Distribution 6.1 Discrete Random Variables 6.2 Binomial Probability Distribution 6.3 Continuous Random Variables and the Normal Probability Distribution 6.1 Discrete

More information

Homework 4 Solutions

Homework 4 Solutions Homework 4 Solutions February 10, 2014 1. Random variables X and Y have the joint PMF { c(x 2 + y 2 ) if x {1, 2, 4} and y {1, 3} p X,Y (x, y) = 0 otherwise (1) (a) What is the value of the constant c?

More information

Random Variables. Consider a probability model (Ω, P ). Discrete Random Variables Chs. 2, 3, 4. Definition. A random variable is a function

Random Variables. Consider a probability model (Ω, P ). Discrete Random Variables Chs. 2, 3, 4. Definition. A random variable is a function Rom Variables Discrete Rom Variables Chs.,, 4 Rom Variables Probability Mass Functions Expectation: The Mean Variance Special Distributions Hypergeometric Binomial Poisson Joint Distributions Independence

More information

Binomial Distribution

Binomial Distribution 1. The experiment consists of a sequence of n smaller experiments called trials, where n is fixed in advance of the experiment; 2. Each trial can result in one of the same two possible outcomes (dichotomous

More information