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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 : Three bets Math 45 Introduction to Probability Lecture 5 Kenneth Harris Department of Mathematics University of Michigan February, 009. A casino offers the following bets (the fairest bets in the casino!) You get $0 (i.e., you can wal away) You get $0 with probability, and otherwise pay $0. You get $0, 000 with probability 0, and otherwise pay $0. All three bets have the same expectation: 0, but they differ in how spread out they are about their mean: This bet is not spread out at all. This bet is symmetrically disposed not too far from its mean. This bet is very spread out. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Definition: Standard Deviation is a measure of the spread of a random variable. The standard deviation is more lie a distance function than variance. Definition Let X be a random variable with distribution p X and mean µ. The variance of X, denoted by Var(X), is defined as Var(X) (r µ) p X (r). The variance is a weighted average of the squared distance from the mean. The variance of X is standardly written σ (X), where σ(x) is often used as a measure of the spread of X. Definition The standard deviation of a random variable X is the positive square root of the variation: SD(X) Var(X). Equivalently, if the mean of X has distribution p X and mean µ, its pmf p(r) then SD(X) (r µ) p X (r). The standard deviation of X is standardly written σ(x). Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, /

2 Three bets revisited Throwing Dice revisted. Our three bets with mean 0: You get $0 (X ). You get $0 with probability, and otherwise pay $0 (X ). You get $0, 000 with probability 0, and otherwise pay $0 (X ). They differ dramatically in the variance: Var(X ) 0 Var(X ) (0 0) + ( 0 0) 00 Var(X ) ( 0 )(0 4 0) + 0 ( 0 0) 0 8 and so too in standard variation, SD(X ) 0 SD(X ) 0 SD(X ) Let X be the outcome of the throw of a single fair die. The distribution of X is So, E[X] Var(X) p X () ( 7 ) SD(X).7 Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Caveat on Caveat on There is a tendancy to believe the standard deviation gives a good measure of dispersion: expected deviation from the mean. However, squaring big deviations tend to dominate the sum. Consider the random variable X with distribution: So, p X () 9 0 p X (0) 0 E[X] Var(X) (.9) (8.) SD(X).7 If deviation from the mean varies alot, then standard deviation is NOT a good measure of the expected deviation. A closer formula might be r µ p X (r). In this case, our previous example gives an average deviation SD(X).7 0 While this version of dispersion of values has good statistical properties, it is much more difficult mathematical properties, so is little used. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, /

3 . A roulette wheel is divided into 8 slotted sectors. 8 are red, 8 are blac and are green, 6 are numbered to 6, together with one mared 0 and one mared 00. A croupier spins the wheel and throws an ivory ball. The success of a bet depends into which slot the ball falls. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Compare the following two bets: Bet $ on red at even money (X). Bet $ on the number 7 at 5 : (Y ). You win $6 on a $ bet. We compare the expected values E[X] ( ) E[Y ] ( ) You can expect to lose about 5 cents a bet in either case. The variance is dramatically different though Var[X] ( + 9 ) ( + 9 ) Var[Y ] ( + 9 ) (5 + 9 ) 8. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / I played both bets 0,000 times. The standard mean of the outcome of the bet and the variance of the outcomes based on the computed mean is as follows: Even money on red: Mean: (E[X] 0.05) : (Var(x) 0.997) 5 : on the number 7: Mean: (E[X] 0.05) :.5 (Var(x).) I placed a 000 bets at the roulette table and computed my total winnings. Here is the mean and variance over 00 plays of each type of bet. Even money on red for 000 bets: Mean winnings: 5.5 (000E[X] 5) : 70.5 (000Var(x) 997) 5 : on the number 7 for 000 bets: Mean: 5.9 (000E[X] 5) : 9, 48 (000Var(x), 0) This suggests that running these independent trials 000 times leads to a 000-fold increase in expectation and variance. This is the case, and we will loo into this relationship in Chapter 7. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, /

4 Linearity properties Proof of Theorem Let X be a random variable with distribution p X and let Y ax + b with distribution p Y. See Ross, Corollary 4. and page 50. Theorem Let a and b be constants. For any random variable X, E[aX + b] ae[x] + b Var(aX + b) a Var(X) Since ar + b ar + b if and only if r r, it follows: p X (r) p Y (ar + b). So, E[aX + b] E[Y ] (ar + b) p Y (ar + b) a r:p Y (ar+b)>0 (ar + b) p X (r) r p X (r) + b p X (r) ae[x] + b. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Proof of Theorem continued : Temperature readings Let µ E[X], so that E[Y ] E[aX + b] a µ + b. Since Var(aX + b) Var(Y ), Var(aX + b) (ar + b (a µ + b)) p Y (ar + b) r:p Y (ar+b)>0 (ar a µ) p X (r) a (r µ) p X (r) a Var(X). In a certain manufacturing process, the (Fahrenheit) temperature never varies more than two degrees from 6 F. The temperature is a random variable X with distribution Find the following temp ( F ) prob temp ( C) Compute E[X], Var(X) and SD(X). It is decided to convert the temperature readings to Celcius, so that Y 5 9 (X ). Compute E[Y ], Var(Y), and SD(Y ). Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, /

5 : Temperature readings In Fahrenheit E[X] Var(X) ( ) 0 + ( ) (0) 0 + () 0 + () SD(X) 5.. In Celcius E[Y ] E[ 5 9 (X )] 5 60 E[X] Var(Y ) Var( 5 9 SD(Y ) 5 0 (X )) Var(X) Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Definition We generalize the previous result to arbitrary functions. Definition Let X be a random variable over sample space S, and g : R R. We write g(x) for the random variable Y on S defined by Y (s) g(x(s)) for all s S. ax + b is the random variable: ax(s) + b for s S, X is the random variable: ( X(s) ) for s S. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Proposition Proof of Proposition See Ross, Proposition 4. Proposition Let X be a random variable with distribution p X. For any real-valued function g, E[g(X)] g(r)p(r). Let X be a random variable with distribution, and let Y g(x) be the random variable with distribution p Y. It is not necessarily true that p Y (g(r)) p X (r), since g may map several numbers to the same value. For example, let X be the random variable with distribution p X ( ) p X (0) p X (). Then Y X is the random variable with distribution p Y (0) p Y (). So, the distributions for X and Y are different. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 /

6 Proof of Proposition Corollary However, the following is true for each real number q: p Y (q) p X (r). r:g(r)q By regrouping the summation: g(r) p X (r) g(r) p X (r) q:p Y (q)>0 r:g(r)q q:p Y (q)>0 q:p Y (q)>0 q r:g(r)q q p Y (q) E[Y ] E[g(X)]. p X (r) The following corollary maes computation of variance much easier. Corollary Let X be a random variable with expected value µ. Then Var(X) E[(X µ) ] E[X ] µ That is, Var(X) E[X ] ( E[X] ) Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Proof of Corollary We defined the variance of a random variable X with distribution p X and expected value µ by Var(X) (r µ) p X (r) E[(X µ) ], by the previous theorem with g(x) (x µ). Var(X) (r µ) p X (r) (r rµ + µ ) p X (r) r p X (r) µ r p X (r) + µ p X (r) E[X ] µ + µ E[X ] µ E[X ] ( E[X] ). : Algebra. Let X be a random variable with E[X] 0 and Var(X) 4. Compute the following (a) E[X ] (b) E[X + 0] and E[ X] (c) Var(X + 0) and Var( X) (d) SD(X), SD(X + 0) and SD( X). Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, /

7 : Algebra : children Given: E[X] 0 and Var(X) 4 (a). Since Var(X) E[X ] ( E[X] ), (b). (c). (c). E[X ] Var(X) + ( E[X] ) E[X + 0] E[X] E[ X] E[X] 0 Var(X + 0) 9Var(X) 6 V ( X) V (X) 4 SD(X) 4 SD(X + 0) 6 SD( X) 4 Find the expected value and the variance for the number of boys and girls in a royal family that has children until there is a boy or until there are three children, whichever comes first. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / continued Solution. Let B (G) be the random variable which counts the boys (girls). Then, the distributions are given by the table: B p B G p G G p G A die is loaded so that the probability of a face coming up is proportional to the number on the face. Find the expected value, variance and standard deviation of the face value. Solution. Let X be the random value of the face value. X has distribution given by E[B] 7 8 E[G] 7 8 E[B ] 7 8 E[G ] 5 8 Var[B] 7 64 Var[G] 7 64 p X () since Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 /

8 continued : heads Compute. p X () Recall, for a fair die these values are 6. E[X] 4. E[X ] 49 Var(X) 4 6. SD(X).46 E[X].5 Var(X).9 SD(X).7 What is the expected number of heads in 4 tosses of a fair coin? What is the standard deviation? Solution. Let X be the random variable counting heads. X has distribution ( ) 4 p X () So, E[X] E[X ] 4 ( ) ( ) Var(X) SD(X) Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, 009 / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / 4: heads 4: heads What is the expected number of heads in 6 tosses of a fair coin? What is the standard deviation? Solution. Let X be the random variable counting heads. X has distribution ( ) 6 p X () So, E[X] E[X ] Var(X) 6 ( ) ( ) SD(X).5 Random variables for counting heads for various numbers of tosses. X E[X] Var(X) We explore this trend further on Monday. Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, /

9 5: Jac and Jill, redoux 5 continued Solution. Let X count the number of rounds played. The probability of a winning throw in any given round is Jac and Jill are playing the game of Heads. Jill s coin is biased of the time heads, Jac s coin is a fair coin. Jac, always the gentleman, allows Jill to toss first. They agree to stop after four rounds. What is the expected number of rounds? What is the variance? +. So, the probability that play goes rounds (when < 4) is p X () ( ) () The last round also has the possibility of no winner: p X (4) ( ) 4 ( ) + ( ) 4 () 4 Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, / 5 continued The probabilities for X are n p X n p X The expected number of rounds of the game: E[X] E[X ] 76 7 Var(X) 0.94 SD(X) 0.97 Kenneth Harris (Math 45) Math 45 Introduction to Probability Lecture 5 February, /

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

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

Statistics and Random Variables. Math 425 Introduction to Probability Lecture 14. Finite valued Random Variables. Expectation defined

Statistics and Random Variables. Math 425 Introduction to Probability Lecture 14. Finite valued Random Variables. Expectation defined Expectation Statistics and Random Variables Math 425 Introduction to Probability Lecture 4 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 9, 2009 When a large

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

Lecture 16: Expected value, variance, independence and Chebyshev inequality

Lecture 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 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

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

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

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

More information

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 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 information

AMS 5 CHANCE VARIABILITY

AMS 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 information

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

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

More information

Elementary Statistics and Inference. Elementary Statistics and Inference. 17 Expected Value and Standard Error. 22S:025 or 7P:025.

Elementary Statistics and Inference. Elementary Statistics and Inference. 17 Expected Value and Standard Error. 22S:025 or 7P:025. Elementary Statistics and Inference S:05 or 7P:05 Lecture Elementary Statistics and Inference S:05 or 7P:05 Chapter 7 A. The Expected Value In a chance process (probability experiment) the outcomes of

More information

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

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

More information

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

Mathematical Expectation

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

More information

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

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

More information

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

13.0 Central Limit Theorem

13.0 Central Limit Theorem 13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated

More information

Chapter 16: law of averages

Chapter 16: law of averages Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

More information

ECE 316 Probability Theory and Random Processes

ECE 316 Probability Theory and Random Processes ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 2) Xinxin Fan Problems 20. A gambling book recommends the following winning strategy for the game of roulette. It recommends that

More information

Statistics 100A Homework 3 Solutions

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

More information

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

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum

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

Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev

Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev Law of Large Numbers Alexandra Barbato and Craig O Connell Honors 391A Mathematical Gems Jenia Tevelev Jacob Bernoulli Life of Jacob Bernoulli Born into a family of important citizens in Basel, Switzerland

More information

Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2

Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2 Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability

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

Lecture 5: Mathematical Expectation

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

More information

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

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

More information

MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010

MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010 MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times

More information

Probability and Statistical Methods. Chapter 4 Mathematical Expectation

Probability and Statistical Methods. Chapter 4 Mathematical Expectation Math 3 Chapter 4 Mathematical Epectation Mean of a Random Variable Definition. Let be a random variable with probability distribution f( ). The mean or epected value of is, f( ) µ = µ = E =, if is a discrete

More information

6.042/18.062J Mathematics for Computer Science. Expected Value I

6.042/18.062J Mathematics for Computer Science. Expected Value I 6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

More information

$2 4 40 + ( $1) = 40

$2 4 40 + ( $1) = 40 THE EXPECTED VALUE FOR THE SUM OF THE DRAWS In the game of Keno there are 80 balls, numbered 1 through 80. On each play, the casino chooses 20 balls at random without replacement. Suppose you bet on the

More information

STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science

STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science Mondays 2:10 4:00 (GB 220) and Wednesdays 2:10 4:00 (various) Jeffrey Rosenthal Professor of Statistics, University of Toronto

More information

3. Continuous Random Variables

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

More information

STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012)

STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012) STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012) TA: Zhen (Alan) Zhang zhangz19@stt.msu.edu Office hour: (C500 WH) 1:45 2:45PM Tuesday (office tel.: 432-3342) Help-room: (A102 WH) 11:20AM-12:30PM,

More information

36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability 36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

More information

3.2 Roulette and Markov Chains

3.2 Roulette and Markov Chains 238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.

More information

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either

More information

Problem sets for BUEC 333 Part 1: Probability and Statistics

Problem sets for BUEC 333 Part 1: Probability and Statistics Problem sets for BUEC 333 Part 1: Probability and Statistics I will indicate the relevant exercises for each week at the end of the Wednesday lecture. Numbered exercises are back-of-chapter exercises from

More information

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

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

More information

Random Variables, Expectation, Distributions

Random 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 information

Econ 132 C. Health Insurance: U.S., Risk Pooling, Risk Aversion, Moral Hazard, Rand Study 7

Econ 132 C. Health Insurance: U.S., Risk Pooling, Risk Aversion, Moral Hazard, Rand Study 7 Econ 132 C. Health Insurance: U.S., Risk Pooling, Risk Aversion, Moral Hazard, Rand Study 7 C2. Health Insurance: Risk Pooling Health insurance works by pooling individuals together to reduce the variability

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

Chapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary.

Chapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary. Overview Box Part V Variability The Averages Box We will look at various chance : Tossing coins, rolling, playing Sampling voters We will use something called s to analyze these. Box s help to translate

More information

We rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is

We rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is Roulette: On an American roulette wheel here are 38 compartments where the ball can land. They are numbered 1-36, and there are two compartments labeled 0 and 00. Half of the compartments numbered 1-36

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

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

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

Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

More information

Lahore University of Management Sciences

Lahore 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

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

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

Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

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

The Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?

The Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going? The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums

More information

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

More information

In the situations that we will encounter, we may generally calculate the probability of an event

In the situations that we will encounter, we may generally calculate the probability of an event What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

More information

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

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

More information

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

Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

More information

The Math. P (x) = 5! = 1 2 3 4 5 = 120.

The Math. P (x) = 5! = 1 2 3 4 5 = 120. The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct

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

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

Statistics 100A Homework 4 Solutions

Statistics 100A Homework 4 Solutions Chapter 4 Statistics 00A Homework 4 Solutions Ryan Rosario 39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn.

More information

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

Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008 Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the

More information

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

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

John Kerrich s coin-tossing Experiment. Law of Averages - pg. 294 Moore s Text

John Kerrich s coin-tossing Experiment. Law of Averages - pg. 294 Moore s Text Law of Averages - pg. 294 Moore s Text 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 the

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

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks 6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

More information

The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES

The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number

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

Elementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.

Elementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025. Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers

More information

Continuous Random Variables

Continuous 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 information

Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

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

Chapter 5. Random variables

Chapter 5. Random variables Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

More information

Solution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections 4.3-4.4) Homework Solutions. Section 4.

Solution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections 4.3-4.4) Homework Solutions. Section 4. Math 115 N. Psomas Chapter 4 (Sections 4.3-4.4) Homework s Section 4.3 4.53 Discrete or continuous. In each of the following situations decide if the random variable is discrete or continuous and give

More information

Notes 11 Autumn 2005

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

More information

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

Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

More information

Thursday, November 13: 6.1 Discrete Random Variables

Thursday, November 13: 6.1 Discrete Random Variables Thursday, November 13: 6.1 Discrete Random Variables Read 347 350 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.

More information

Probability and Expected Value

Probability and Expected Value Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

More information

The Casino Lab STATION 1: CRAPS

The Casino Lab STATION 1: CRAPS The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

More information

arxiv:1112.0829v1 [math.pr] 5 Dec 2011

arxiv:1112.0829v1 [math.pr] 5 Dec 2011 How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

More information

Statistics 100A Homework 4 Solutions

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

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

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

Expected Value. 24 February 2014. Expected Value 24 February 2014 1/19

Expected Value. 24 February 2014. Expected Value 24 February 2014 1/19 Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery

More information

HONORS STATISTICS. Mrs. Garrett Block 2 & 3

HONORS STATISTICS. Mrs. Garrett Block 2 & 3 HONORS STATISTICS Mrs. Garrett Block 2 & 3 Tuesday December 4, 2012 1 Daily Agenda 1. Welcome to class 2. Please find folder and take your seat. 3. Review OTL C7#1 4. Notes and practice 7.2 day 1 5. Folders

More information

Chapter 13 & 14 - Probability PART

Chapter 13 & 14 - Probability PART Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph

More information

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

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

More information

X: 0 1 2 3 4 5 6 7 8 9 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001

X: 0 1 2 3 4 5 6 7 8 9 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001 Tuesday, January 17: 6.1 Discrete Random Variables Read 341 344 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.

More information

Probability and Statistics

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

More information

MULTIVARIATE PROBABILITY DISTRIBUTIONS

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

More information

Fourth Problem Assignment

Fourth Problem Assignment EECS 401 Due on Feb 2, 2007 PROBLEM 1 (25 points) Joe and Helen each know that the a priori probability that her mother will be home on any given night is 0.6. However, Helen can determine her mother s

More information

10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation 10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

More information

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k. REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

More information

BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

More information