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


 Mervin Skinner
 2 years ago
 Views:
Transcription
1 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 Standard deviation Covariance and correlation Covariance is the average of (X µ X )(Y µ Y ) Correlation is a scaled version of covariance Lots of examples PHP 2510 Oct 8,
2 Expected value Synonyms for expected value: average, mean The expectation or expected value of a random variable X is a weighted average of its possible outcomes. For a discrete random variable, each outcome is weighted by its probability of occurrence, using the mass function: E(X) = i x i P (X = x i ) = i x i p(x i ) For a continuous random variable, each outcome is weighted by the relative frequency of its occurrence, using the density function: E(X) = x f(x) dx PHP 2510 Oct 8,
3 Examples: Discrete random variables Example 1. Let X denote the number of boys in a family with three children. Assume the probability of having a boy is.5. Step 1: Compute the mass function k p(k) Step 2: Compute weighted average E(X) = 3 k p(k) k=0 = (0)(.125) + (1)(.375) + (2)(.375) + (3)(.125) = 1.5 PHP 2510 Oct 8,
4 Example 2: Roulette. In roulette, a ball is tossed on a spinning wheel, and it lands on one of 38 numbers (each of 1 to 36, plus 0 and 00). If you bet $1 on a particular number, the payoff for winning is $36. Suppose you bet $1 on the number 12. Define the random variable X to be your winnings on one play of the roulette wheel. Then 36 if the number is 12 X = 1 if the number is not 12 Find E(X), or your expected winnings. PHP 2510 Oct 8,
5 Step 1: Compute mass function k p(k) Step 2: Compute E(X) as weighted average of outcomes E(X) = k p(k) k= 1,36 ( ) 37 = ( 1) + (36) 38 = ( ) 1 38 Question: What the expected return in 100 plays of roulette? PHP 2510 Oct 8,
6 Expected value for common discrete RV s Binomial. If X has the binomial distribution with parameters n and π, then E(X) = nπ. Example: Toss a coin 50 times, and let X denote the number of heads. Then E(X) = nπ = 50.5 = 25 Example: The proportion of individuals with coronary artery disease is.3. In a sample of 45 individuals, what is the expected number of cases of CAD? E(X) = nπ = 45.3 = 13.5 Suppose one person is selected from the population. Define a random variable Y such that Y = 1 if the person has CAD and Y = 0 if not. Then E(Y ) = nπ = 1.3 =.3 PHP 2510 Oct 8,
7 Poisson. If X has the Poisson distribution with rate parameter λ, then E(X) = λ. This is because ) λ λk E(X) = k (e = λ k! k=0 The mean of a Poisson RV is the number of events you expect to observe. PHP 2510 Oct 8,
8 Geometric. If X has the Geometric distribution with success probability π, then E(X) = 1/π. This is because E(X) = k { (1 π) k 1 π } = 1 π k=1 The mean of a geometric RV is the number of trials you expect to require before observing the first success. Hence if the success probability π is low, E(X) will be high; and viceversa. Example. If you roll two dice, the probability of rolling a 3 is 2/36 or about Let X denote the number of rolls until a 3 comes up. What is E(X)? (Ans: 18) PHP 2510 Oct 8,
9 Expected value for continuous RV Let X be a continuous random variable defined on an interval A. Then the expected value is a weighted average of outcomes, weighted by the relative frequency of each outcome. The weighted average is computed using an integral, E(X) = x f(x) dx A PHP 2510 Oct 8,
10 Example. Suppose X is a uniform random variable on the interval [1, 4]. Find E(X). Step 1: Recall that f(x) = = 1 3, and that the interval A is [1, 4]. So the appropriate integral is 4 1 x f(x) dx = 4 1 x 1 3 dx Step 2: Evaluate the integral 4 1 x 1 3 dx = 1 3 x = 2.5 PHP 2510 Oct 8,
11 Expected values for common continuous RV s Normal. If X has a normal distribution with parameters µ and σ, then E(X) = µ. Exponential. If X has the exponential distribution with parameter θ, then E(X) = θ. In this case, θ is the expected waiting time until an event occurs, and 1/θ is called the event rate. PHP 2510 Oct 8,
12 Some properties of expected values. 1. Linear combinations. If a and b are constants, then E(aX + b) = ae(x) + b 2. Sums of random variables. The expected value of a sum of random variables is the sum of expected values. E(X 1 + X X n ) = E(X 1 ) + E(X 2 ) + + E(X n ) PHP 2510 Oct 8,
13 Example. Suppose X is a Poisson random variable denoting the number of lottery winners per week. Its expected value is E(X) = 2. What is the expected number of winners over 4 weeks? E(4X) = 4 E(X) = 4 2 = 8 Example. Let X denote the daily low temperature for each day in September, and let E(X) denote its average. Suppose E(X) = 65, measured in degrees Fahrenheit. What is the mean temperature in degrees Celsius? To convert X from F to C, define a new random variable Y = 5 9 X Then using the rule about linear combinations, E(Y ) = E(X) PHP 2510 Oct 8,
14 Computing means from a sample of data Loosely speaking, for a sample of observed data x 1, x 2,..., x n, each of the individual x i can be thought of as having associated probability mass p(x i ) = 1/n. So the sample mean is x = = = 1 n n x i p(x i ) i=1 n x i (1/n) Simply put, take the sum of the observations and divide by n. i=1 Sample means are not expected values! They are random variables. n i=1 We will discuss sample means later on... PHP 2510 Oct 8, x i
15 Variance of a random variables Variance measures dispersion of a random variable s distribution. It is just an average. It is the average squared deviation of a random variable from its mean. To make notation simple, let µ = E(X). Then var(x) = E{(X µ) 2 } In other words, it is the average value of (X µ) 2. For a discrete random variable, var(x) = i (x i µ) 2 p(x i ) For a continuous random variable, var(x) = (x µ) 2 f(x) dx PHP 2510 Oct 8,
16 Example 1 (consumers of alcohol). In a certain population, the proportion of those consuming alcohol is.65. Select a person at random, with X = 1 if consumer of alcohol and X = 0 if not. In this example, E(X) = µ = var(x) = E{(X 0.65) 2 } = i (x i 0.65) 2 p(x i ) = (1 0.65) 2 (0.65) + (0 0.65) 2 (0.35) =.228 Example 2. Suppose instead the probability was 0.1. What then is var(x)? Ans = Pattern: For a Binomial random variable X with n = 1 and success probability π, var(x) = π(1 π) PHP 2510 Oct 8,
17 Properties of variance If a and b are constants, then var(ax + b) = a 2 var(x) (Why is b not included?) If X 1, X 2,..., X n are independent random variables, then var(x 1 + X X n ) = var(x 1 ) + var(x 2 ) + + var(x n ) PHP 2510 Oct 8,
18 Computing variances from a sample of data Like with the sample mean, for a sample of observed data x 1, x 2,..., x n, each of the individual x i can be thought of as having associated probability mass p(x i ) = 1/n. To calculate the sample variance, we take an average of (x i x) 2. The sample variance is S 2 = = = 1 n n (x i x) 2 p(x i ) i=1 n (x i x) 2 (1/n) i=1 n (x i x) 2 i=1 1 It is more common to use n 1 instead of 1 n. We will discuss reasons for this later. For now, you should think of variance as an average. PHP 2510 Oct 8,
19 Standard deviation The standard deviation measures the average distance of a random variable X from its mean. By definition, SD(X) = var(x). The logic goes like this: 1. because var(x) measures average squared deviation between X and its mean; and 2. because SD(X) = var(x); then 3. SD(X) is approximately equal to the average absolute deviation between X and its mean PHP 2510 Oct 8,
20 Example. In September in Providence, noon time temperature has mean 65 and variance 100. What is the SD of the temperatures? Select a day at random. What does SD tell us about the temperature on that day, relative to the average temperature? Suppose noon time temps are normally distributed. Should a noon time temperature of 85 be considered unusual? Why or why not? PHP 2510 Oct 8,
21 Mean and variance for some common RV s Random variable Mass or Density Function E(X) var(x) Binomial(n, π) ( n ) x π x (1 π) n x nπ nπ(1 π) Poisson(λ) e λ λ x /x! λ λ Geometric(π) (1 π) x 1 π 1/π 1/π 2 Normal(µ, σ 2 ) µ σ 2 Exponential(θ) (1/θ)e θ/x 1/θ 1/θ 2 PHP 2510 Oct 8,
22 Correlation and Covariance Correlation and covariance are one way to measure association between two random variables that are observed at the same time on the same unit. Example: Height and weight measured on the same person Example: years of education and income Example: two successive measures of weight, taken on the same person but one year apart. PHP 2510 Oct 8,
23 Covariance Covariance measures the degree to which two variables differ from their mean. It is an average: cov(x, Y ) = E {(X µ X )(Y µ Y )} cov(x, Y ) > 0 means that X and Y tend to vary in the same direction relative to their means (both higher or both lower). They have a positive association. Example: height and weight cov(x, Y ) < 0 means that X and Y tend to vary in opposite directions relative to their means (when one is higher, the other is lower). They have a negative association. Example: weight and minutes of exercise per day cov(x, Y ) = 0 generally means that X and Y are not associated. PHP 2510 Oct 8,
24 Example: mean arterial pressure and body mass index during pregnancy SUMMARY STATISTICS Variable Obs Mean Std. Dev map bmi Give an interpretation for SD here. PHP 2510 Oct 8,
25 map bmi PHP 2510 Oct 8,
26 Computing covariance For individual i, let m i denote MAP and let b i denote BMI. In this table, prod represents (m i m) (b i b) Recall m = 76.6 and b = To compute covariance, we take the average (sample mean) of the products (following pages) DATA EXCERPT map24 (m_i) bmi (b_i) prod PHP 2510 Oct 8,
27 SUMMARY STATISTICS Variable Obs Mean prod PHP 2510 Oct 8,
28 Computing covariance from a sample Like mean and variance, covariance is an average. In a sample of pairs (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ), we can assume each pair is observed with probability p(x i, y i ) = 1/n. Then the sample covariance is a weighted average of (x i x) (y i y): n ĉov(x, Y ) = (x i x) (y i y) p(x i, y i ) = 1 n i=1 n (x i x) (y i y) i=1 PHP 2510 Oct 8,
29 Correlation is a standardized covariance corr(x, Y ) = Always between 1 and 1 cov(x, Y ) SD(X) SD(Y ) Measures degree of linear relationship (If relationship not linear, correlation not an appropriate measure of association) Pearson s sample correlation plugs in sample estimates for the quantities in the formula above ĉorr(x, Y ) = (1/n) n i=1 (x i x)(y i y) S x S y PHP 2510 Oct 8,
30 SUMMARY STATISTICS Variable Obs Mean Std. Dev. Min Max prod map bmi CORRELATION COEFFICIENT (obs=326) bmi map Using the numbers on the table above, how would you obtain the correlation coefficient? PHP 2510 Oct 8,
Expected Value. Let X be a discrete random variable which takes values in S X = {x 1, x 2,..., x n }
Expected Value Let X be a discrete random variable which takes values in S X = {x 1, x 2,..., x n } Expected Value or Mean of X: E(X) = n x i p(x i ) i=1 Example: Roll one die Let X be outcome of rolling
More informationExample. 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 informationUniversity 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 informationRandom Variables, Expectation, Distributions
Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability
More informationJoint 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 informationRandom 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 informationBivariate Distributions
Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is
More informationExamination 110 Probability and Statistics Examination
Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiplechoice test questions. The test is a threehour examination
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More information2.8 Expected values and variance
y 1 b a 0 y 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 a b (a) Probability density function x 0 a b (b) Cumulative distribution function x Figure 2.3: The probability density function and cumulative distribution function
More informationMath 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 informationReview. Lecture 3: Probability Distributions. Poisson Distribution. May 8, 2012 GENOME 560, Spring Su In Lee, CSE & GS
Lecture 3: Probability Distributions May 8, 202 GENOME 560, Spring 202 Su In Lee, CSE & GS suinlee@uw.edu Review Random variables Discrete: Probability mass function (pmf) Continuous: Probability density
More informationThe basics of probability theory. Distribution of variables, some important distributions
The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a
More informationChapter 6 Random Variables
Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:
More informationST 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 informationFor 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 (19031987) 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 informationJoint 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 informationEE 322: Probabilistic Methods for Electrical Engineers. Zhengdao Wang Department of Electrical and Computer Engineering Iowa State University
EE 322: Probabilistic Methods for Electrical Engineers Zhengdao Wang Department of Electrical and Computer Engineering Iowa State University Discrete Random Variables 1 Introduction to Random Variables
More informationChapter 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 informationSummary of Probability
Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible
More informationPHP 2510 Central limit theorem, confidence intervals. PHP 2510 October 20,
PHP 2510 Central limit theorem, confidence intervals PHP 2510 October 20, 2008 1 Distribution of the sample mean Case 1: Population distribution is normal For an individual in the population, X i N(µ,
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 101634501 Probability and Statistics for Engineers Winter 20102011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete
More informationContinuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.
UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Neda Farzinnia, UCLA Statistics University of California,
More informationPOL 571: Expectation and Functions of Random Variables
POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior
More informationContinuous Random Variables
Continuous Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Continuous Random Variables 2 11 Introduction 2 12 Probability Density Functions 3 13 Transformations 5 2 Mean, Variance and Quantiles
More informationRandom 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 informationDefinition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).
Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The
More informationProbability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0
Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall
More informationDISCRETE 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 informationDiscrete 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 informationTopic 2: Scalar random variables. Definition of random variables
Topic 2: Scalar random variables Discrete and continuous random variables Probability distribution and densities (cdf, pmf, pdf) Important random variables Expectation, mean, variance, moments Markov and
More informationMATH 201. Final ANSWERS August 12, 2016
MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6sided dice, five 8sided dice, and six 0sided dice. A die is drawn from the bag and then rolled.
More information3. Continuous Random Variables
3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such
More informationMath 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 informationRandom 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 informationJoint Distribution and Correlation
Joint Distribution and Correlation Michael Ash Lecture 3 Reminder: Start working on the Problem Set Mean and Variance of Linear Functions of an R.V. Linear Function of an R.V. Y = a + bx What are the properties
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationContinuous Distributions
MAT 2379 3X (Summer 2012) Continuous Distributions Up to now we have been working with discrete random variables whose R X is finite or countable. However we will have to allow for variables that can take
More informationWill Landau. Feb 26, 2013
,, and,, and Iowa State University Feb 26, 213 Iowa State University Feb 26, 213 1 / 27 Outline,, and Iowa State University Feb 26, 213 2 / 27 of continuous distributions The pquantile of a random variable,
More informationMATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS
MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution
More information3 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 informationProperties 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 informationLecture 18 Chapter 6: Empirical Statistics
Lecture 18 Chapter 6: Empirical Statistics M. George Akritas Definition (Sample Covariance and Pearson s Correlation) Let (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. The sample
More informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More information9740/02 October/November MATHEMATICS (H2) Paper 2 Suggested Solutions. Ensure calculator s mode is in Radians. (iii) Refer to part (i)
GCE Level October/November 8 Suggested Solutions Mathematics H (974/) version. MTHEMTICS (H) Paper Suggested Solutions. Topic : Functions and Graphs (i) 974/ October/November 8 (iii) (iv) f(x) g(x)
More informationMath 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 informationContinuous random variables
Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the socalled continuous random variables. A random
More information5. 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 informationP (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 informationCovariance and Correlation
Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a databased relative frequency distribution by measures of location and spread, such
More informationRandom 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 informationJointly 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 informationExercises in Probability Theory Nikolai Chernov
Exercises in Probability Theory Nikolai Chernov All exercises (except Chapters 16 and 17) are taken from two books: R. Durrett, The Essentials of Probability, Duxbury Press, 1994 S. Ghahramani, Fundamentals
More informationIntroduction to Probability
Motoya Machida January 7, 6 This material is designed to provide a foundation in the mathematics of probability. We begin with the basic concepts of probability, such as events, random variables, independence,
More informationReview 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 information1. Consider an untested batch of memory chips that have a known failure rate of 8% (yield = 92%).
eview of Introduction to Probability and Statistics Chris Mack, http://www.lithoguru.com/scientist/statistics/review.html omework #2 Solutions 1. Consider an untested batch of memory chips that have a
More informationContinuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4  and Cengage
4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4  and Cengage Continuous r.v. A random variable X is continuous if possible values
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationRandom 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 informationThe Expected Value of X
3.3 Expected Values The Expected Value of X Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X) or μ X or just μ, is 2 Example
More informationSome continuous and discrete distributions
Some continuous and discrete distributions Table of contents I. Continuous distributions and transformation rules. A. Standard uniform distribution U[0, 1]. B. Uniform distribution U[a, b]. C. Standard
More informationChapters 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 informationNormal 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 1.1.23, 2.1 Normal distribution Math 283 / October
More informationMA 1125 Lecture 14  Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4  Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationMath 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 informationMATH 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 informationST 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 informationChapter 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 37, 38 The remaining discrete random
More informationConditional expectation
A Conditional expectation A.1 Review of conditional densities, expectations We start with the continuous case. This is sections 6.6 and 6.8 in the book. Let X, Y be continuous random variables. We defined
More information4. 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 informationMath 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 informationFALL 2005 EXAM C SOLUTIONS
FALL 005 EXAM C SOLUTIONS Question #1 Key: D S ˆ(300) = 3/10 (there are three observations greater than 300) H ˆ (300) = ln[ S ˆ (300)] = ln(0.3) = 1.0. Question # EX ( λ) = VarX ( λ) = λ µ = v = E( λ)
More informationLecture 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 informationTopic 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 informationPractice Exam 1: Long List 18.05, Spring 2014
Practice Eam : Long List 8.05, Spring 204 Counting and Probability. A full house in poker is a hand where three cards share one rank and two cards share another rank. How many ways are there to get a fullhouse?
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
More information5. Conditional Expected Value
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 5. Conditional Expected Value As usual, our starting point is a random experiment with probability measure P on a sample space Ω. Suppose that X is
More informationMath 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 informationChapter 4 Expected Values
Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges
More informationRANDOM 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 informationContinuous Random Variables
Probability 2  Notes 7 Continuous Random Variables Definition. A random variable X is said to be a continuous random variable if there is a function f X (x) (the probability density function or p.d.f.)
More informationUniversity of California, Berkeley, Statistics 134: Concepts of Probability
University of California, Berkeley, Statistics 134: Concepts of Probability Michael Lugo, Spring 211 Exam 2 solutions 1. A fair twentysided die has its faces labeled 1, 2, 3,..., 2. The die is rolled
More informationL10: Probability, statistics, and estimation theory
L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian
More informationLahore University of Management Sciences
Lahore University of Management Sciences CMPE 501: Applied Probability (Fall 2010) Homework 3: Solution 1. A candy factory has an endless supply of red, orange, yellow, green, blue and violet jelly beans.
More information, where f(x,y) is the joint pdf of X and Y. Therefore
INDEPENDENCE, COVARIANCE AND CORRELATION M384G/374G Independence: For random variables and, the intuitive idea behind " is independent of " is that the distribution of shouldn't depend on what is. This
More informationCommon 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 informationRenewal Theory. (iv) For s < t, N(t) N(s) equals the number of events in (s, t].
Renewal Theory Def. A stochastic process {N(t), t 0} is said to be a counting process if N(t) represents the total number of events that have occurred up to time t. X 1, X 2,... times between the events
More informationWe 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 informationChapter 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 information3.6: General Hypothesis Tests
3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.
More informationProbability distributions
Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.142.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,
More informationChapter 10: Introducing Probability
Chapter 10: Introducing Probability Randomness and Probability So far, in the first half of the course, we have learned how to examine and obtain data. Now we turn to another very important aspect of Statistics
More informationStatistics  Written Examination MEC Students  BOVISA
Statistics  Written Examination MEC Students  BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.
More informationMeasurements 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 information2. Discrete random variables
2. Discrete random variables Statistics and probability: 21 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 informationAnNajah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)
AnNajah 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 informationMT426 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 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More informationChapter 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 35, 36 Special discrete random variable distributions we will cover
More informationLecture 16: Expected value, variance, independence and Chebyshev inequality
Lecture 16: Expected value, variance, independence and Chebyshev inequality Expected value, variance, and Chebyshev inequality. If X is a random variable recall that the expected value of X, E[X] is the
More informationContents. 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