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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

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

2 History The early history of probability theory is intimately related to questions arising in gambling. For example:

3 History The early history of probability theory is intimately related to questions arising in gambling. For example: Two gamblers playing a game that ends when one has won all of the other s money have to quit in the middle of the game. What is the fair division of the pot?

4 History The early history of probability theory is intimately related to questions arising in gambling. For example: Two gamblers playing a game that ends when one has won all of the other s money have to quit in the middle of the game. What is the fair division of the pot? What is the fair price of a lottery ticket?

6 Expected Value Definition: Let X be a random variable with sample space with corresponding probabilities Ω = {x 0, x 1, x 2,...} P(X = x 0 ) = p 0, P(X = x 1 ) = p 1, P(X = x 2 ) = p 2,... Then the expected value of X, denoted E(X) is E(X) = x 0 p 0 + x 1 p 1 + x 2 p 2... Expected value is just the weighted average of the set of possible outcomes, with weights equal to the probability each outcome occurs.

7 Example: Y Bernoulli(p) Bernoulli Trials: Each trial results in either a 1 or a 0. P(Y = 1) = p P(Y = 0) = (1 p) = q

8 Example: Y Bernoulli(p) Bernoulli Trials: Each trial results in either a 1 or a 0. P(Y = 1) = p P(Y = 0) = (1 p) = q Expected Value: E(Y ) = (0 q) + (1 p) = p

9 Die Rolls: Discrete Uniform Let X be the outcome of the roll of a fair die: Ω = {1, 2, 3, 4, 5, 6} Fair: each value occurs with probability 1/6. E(X) = (1 1 6 ) + (2 1 6 ) + (3 1 6 ) + (4 1 6 ) + (5 1 6 ) + (6 1 6 ) Adding it up: E(X) = 21 6 = 3.5

10 Example: X is Geometric(p) Probabilities: For k = 0, 1, 2,... P(X = k) = p q k

11 Example: X is Geometric(p) Probabilities: For k = 0, 1, 2,... P(X = k) = p q k Expectation by Definition: E(X) = (0 p q 0 ) + (1 p q 1 ) + (2 p q 2 ) +...

12 Example: X is Geometric(p) Probabilities: For k = 0, 1, 2,... P(X = k) = p q k Expectation by Definition: E(X) = (0 p q 0 ) + (1 p q 1 ) + (2 p q 2 ) +... A Math 112 exercise in summing an infinite series.

13 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six.

14 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six. Consider how long we wait for Heads when tossing a fair coin. In the long run, roughly 1 of every 2 tosses yields Heads. Thus the long run average is 1 Tail before each Heads.

15 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six. Consider how long we wait for Heads when tossing a fair coin. In the long run, roughly 1 of every 2 tosses yields Heads. Thus the long run average is 1 Tail before each Heads. 5 = 6 1 = 1 1/6 1 = 5/6 1/6 1 = 2 1 = 1 1/2 1 = 1/2 1/2

16 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six. Consider how long we wait for Heads when tossing a fair coin. In the long run, roughly 1 of every 2 tosses yields Heads. Thus the long run average is 1 Tail before each Heads. 5 = 6 1 = 1 1/6 1 = 5/6 1/6 1 = 2 1 = 1 1/2 1 = 1/2 1/2 1 p 1 = 1 p p = q p

17 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution.

18 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average...

19 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average... Gambling: The fair price of a wager.

20 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average... Gambling: The fair price of a wager. Average as typical value: The average family has 2.4 children.

21 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average... Gambling: The fair price of a wager. Average as typical value: The average family has 2.4 children. Silly question? How many families have 2.4 children?

22 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average... Gambling: The fair price of a wager. Average as typical value: The average family has 2.4 children. Silly question? How many families have 2.4 children? Synonyms: average, mean, expectation value

23 Median: another measure of location Definition: Median Let X be a RV, then any number m satisfying and P(X m) 1 2 P(X m) 1 2 is a median. The median of a distribution may not be unique.

24 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10}

25 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10} 3 satisfies the definition: at least half the population is less than or equal to 3, and at least half is greater than or equal to 3.

26 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10} 3 satisfies the definition: at least half the population is less than or equal to 3, and at least half is greater than or equal to 3. Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 10}

27 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10} 3 satisfies the definition: at least half the population is less than or equal to 3, and at least half is greater than or equal to 3. Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 10} Any number in the interval [2, 3]!

28 Expectation as Typical Value For a symmetric, unimodal population, the mean and median agree, and both seem typical: Density Z

29 Expectation as Typical Value The mean is not so typical for a multimodal or skewed population: Density median mean X

30 Expectation as Typical Value: Income US Household Income: According to the US Census Bureau, in 2009 the median household income in the United States was \$49,777, while the mean household income \$67,976. Roughly 2/3 of all households earn less than the mean household income. The US has a very skewed income distribution. Depending on how you define income, The top 1% of the US population gets about 24% of all income, and the top 10% gets close to 50% of the total.

31 Properties of Expected Value Back to probability Theory! Let X be a RV, and a and b constants, then E(a + b X) = a + b E(X) Like any average, expected values are well behaved with respect to translation (shift) and rescaling.

32 Example Let X be a Bernoulli(1/2) trial. We know E(X) = p = 1/2. Let Y = 2 X 1. What is E(Y )?

33 Example Let X be a Bernoulli(1/2) trial. We know E(X) = p = 1/2. Let Y = 2 X 1. What is E(Y )? From the definition: Ω y = {2 1 1, 2 0 1} = {1, 1}, each with probability 1/2, so E(Y ) = = 0

34 Example Let X be a Bernoulli(1/2) trial. We know E(X) = p = 1/2. Let Y = 2 X 1. What is E(Y )? From the definition: Ω y = {2 1 1, 2 0 1} = {1, 1}, each with probability 1/2, so Using the linearity of E: E(Y ) = = 0 E(Y ) = E(2X 1) = 2E(X) 1 = = 0

35 Expected Values scale like averages The average height of an adult female in the US is about 64 inches. What is the average height of an adult female in the US in centimeters? The conversion factor is 2.54 cm per inch. Thus the average height of an adult female in the US is 2.54 cm in 64 in cm

36 More Properties of Expected Value E is a linear operator Let X and Y be RV s, and a and b constants, then E(a X + b Y ) = a E(X) + b E(Y ) Thus the expected value of a sum is the sum of the expected values: E(X + Y ) = E(X) + E(Y )

37 Expected Value of a Binomial Let X be a Binomial(n,p) RV. From the definiton of expectation, E(X) = k=n k=0 k ( ) n p k q n k k which may be algebraically challenging for some. But we can think of X as the sum of n (independent) Bernoulli(p) trials Y i, each with E(Y i ) = p, so E(X) = E(Y Y n ) = E(Y 1 ) +... E(Y n ) = np

38 Example: Coin Tossing Toss a fair coin 100 times, and count the number of Heads. What is the expected number of Heads?

39 The Poisson Let X Poisson(µ). What is E(X)? Hint: X is like a Binomial(n,p) with large n and small p, where we define µ = n p.

40 Summary Expected value: a weighted average of possible outcomes. Linearity properties: E(X) = x i p i E(a + bx + cy ) = a + be(x) + ce(y ) Expected value of X Binomial(n, p) is the sum of expected values of n Bernoulli(p) trials: E(X) = np

### 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

### 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

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

74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems 5.1.1 Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k

### Math 141. Lecture 3: The Binomial Distribution. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141 Lecture 3: The Binomial Distribution Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Coin Tossing Coin Tosses Independent Coin Tosses Crucial Features

### 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

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

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

### Math 461 Fall 2006 Test 2 Solutions

Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### Practice Problems #4

Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice

### 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

### 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

### 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

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

### MAS108 Probability I

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

### 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

### 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.

### 6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS Certain derivations presented in this course have been somewhat heavy on algebra. For example, determining the expectation of the Binomial distribution (page 5.1 turned

### 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

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

Introduction. ommon probability distributionsi Math 7/8 Probability and Statistics Prof. D. Joyce, 06 I summarize here some of the more common distributions used in probability and statistics. Some are

### 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 ()

### 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

### 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

### 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,

### Continuous 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,

### Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26 Today s Lecture Today s Lecture Where this Fits central tendency/location

### LECTURE 16. Readings: Section 5.1. Lecture outline. Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process

LECTURE 16 Readings: Section 5.1 Lecture outline Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process Number of successes Distribution of interarrival times The

### 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

### 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

### 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

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli

### 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

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

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### Probability distributions

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

### Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

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

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

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

### 2WB05 Simulation Lecture 8: Generating random variables

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

### Unit 4 The Bernoulli and Binomial Distributions

PubHlth 540 4. Bernoulli and Binomial Page 1 of 19 Unit 4 The Bernoulli and Binomial Distributions Topic 1. Review What is a Discrete Probability Distribution... 2. Statistical Expectation.. 3. The Population

### 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.

### 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

### 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

### 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 <

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

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

Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

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

4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

### Section 5-3 Binomial Probability Distributions

Section 5-3 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial

### Feb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179)

Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172-179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities

### 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.

### 6.2. Discrete Probability Distributions

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

### Statistics - 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.

### 1.1 Introduction, and Review of Probability Theory... 3. 1.1.1 Random Variable, Range, Types of Random Variables... 3. 1.1.2 CDF, PDF, Quantiles...

MATH4427 Notebook 1 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 1 MATH4427 Notebook 1 3 1.1 Introduction, and Review of Probability

### 4. Continuous Random Variables, the Pareto and Normal Distributions

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

### The Binomial Distribution. Summer 2003

The Binomial Distribution Summer 2003 Internet Bubble Several industry experts believe that 30% of internet companies will run out of cash in 6 months and that these companies will find it very hard to

### 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

### Mind on Statistics. Chapter 8

Mind on Statistics Chapter 8 Sections 8.1-8.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable

### Bivariate 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

### Practice problems for Homework 11 - Point Estimation

Practice problems for Homework 11 - Point Estimation 1. (10 marks) Suppose we want to select a random sample of size 5 from the current CS 3341 students. Which of the following strategies is the best:

### 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

### University 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 twenty-sided die has its faces labeled 1, 2, 3,..., 2. The die is rolled

### AP Statistics 7!3! 6!

Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

### Characteristics of Binomial Distributions

Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

### 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,

### 2. Discrete random variables

2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be

### 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

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

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

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

ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in

### 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

### 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

### 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

### 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 =

### Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions

Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

### Probability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X

Week 6 notes : Continuous random variables and their probability densities WEEK 6 page 1 uniform, normal, gamma, exponential,chi-squared distributions, normal approx'n to the binomial Uniform [,1] random

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Chapter 2, part 2. Petter Mostad

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

### Expected values, standard errors, Central Limit Theorem. Statistical inference

Expected values, standard errors, Central Limit Theorem FPP 16-18 Statistical inference Up to this point we have focused primarily on exploratory statistical analysis We know dive into the realm of statistical

### 16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of

### Final Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin

Final Mathematics 51, Section 1, Fall 24 Instructor: D.A. Levin Name YOU MUST SHOW YOUR WORK TO RECEIVE CREDIT. A CORRECT ANSWER WITHOUT SHOWING YOUR REASONING WILL NOT RECEIVE CREDIT. Problem Points Possible

### Section 5 Part 2. Probability Distributions for Discrete Random Variables

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

### Models for Discrete Variables

Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

### 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

### 0 x = 0.30 x = 1.10 x = 3.05 x = 4.15 x = 6 0.4 x = 12. f(x) =

. A mail-order computer business has si telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. 0 2 3 4 5 6 p(.0.5.20.25.20.06.04

### 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

### Introduction to Probability

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

### Math/Stats 342: Solutions to Homework

Math/Stats 342: Solutions to Homework Steven Miller (sjm1@williams.edu) November 17, 2011 Abstract Below are solutions / sketches of solutions to the homework problems from Math/Stats 342: Probability

### 6.1. Construct and Interpret Binomial Distributions. p Study probability distributions. Goal VOCABULARY. Your Notes.

6.1 Georgia Performance Standard(s) MM3D1 Your Notes Construct and Interpret Binomial Distributions Goal p Study probability distributions. VOCABULARY Random variable Discrete random variable Continuous

### TEACHER NOTES MATH NSPIRED

Math Objectives Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 p) are sufficiently large. Students will understand that when

### 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

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

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

### The Kelly Betting System for Favorable Games.

The Kelly Betting System for Favorable Games. Thomas Ferguson, Statistics Department, UCLA A Simple Example. Suppose that each day you are offered a gamble with probability 2/3 of winning and probability

### Examination 110 Probability and Statistics Examination

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

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

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