Discrete Random Variables (1) Solutions
|
|
- Grant Peters
- 6 years ago
- Views:
Transcription
1 STAT/MATH 394 A - PROBABILITY I UW Autumn Quarter 06 Néhémy Lim Discrete Random Variables ( Solutions Problem. The probability mass function p X of some discrete real-valued random variable X is given by the following table : (a Give the missing value p X (. x p X (x Since p X is a probability mass function, we have that 4 x=0 p X(x =. Therefore p X ( = (p X (0 + p X ( + p X (3 + p X (4 = ( = 0.. (b Draw the histogram of p X px(x Figure : Histogram of p X (c Give the cumulative distribution function F X of X. By definition, for any real number x, F X (x = P(X x. F X (x = y x p X (y
2 For instance, F X (. = y. p X(y = p X (0 + p X ( + p X ( = 0.7 In our case, X is a discrete random variable, so its cumulative distribution function F X is piecewise constant: 0 x < x < 0.5 x < F X (x = 0.7 x < x < 4 x 4 (d Graph F X..5 F X (x x 0.5 Problem. In each case, determine the constant c so that the following functions p satisfy the conditions of being valid probability mass functions : (a For x X(Ω = {.5,,.5}, p(x = x3 c The first condition requires that p(x 0. For x =.5, p(.5 =.5 3 /c therefore c should be negative. Now if we take x =, we have that p( = 3 /c, which implies that c should be positive. We reach a contradiction. So there is no constant c for p to be a valid probability mass function. (b For x X(Ω = {n N, n }, p(x = 3 ( x c
3 The first condition requires that p(x 0 for all x N, n, which implies that c should be positive. The second condition is that p should sum to. Remember that if (u n is a geometric sequence, with first term u 0 = a and common ratio r, that is : u n+ = ru n with < r <, then the geometric series n 0 u n converges and its sum is : n=0 u n = a/( r. Here, the sequence defined by : u 0 = /c and u n+ = (/c u n is a geometric sequence. Therefore, if < /c <, we have that : p(x = 3 ( x = c x= x= 3 /c /c = 3 c = 3 c = c = 3 c = 5 We must verify that c > 0 and that < /c <, which is the case for c = 5. (c For x X(Ω = N, p(x = c x x! The first condition requires that p(x 0 for all x N, which implies that c should be nonnegative since x /x! > 0. The second condition is that p should sum to. Here, you recognize the exponential series, that is, for any real number a, n=0 an /n! = e a. Therefore, we have that : x=0 It is obvious that c 0. x p(x = c x! = x=0 c e = c = e = e 3
4 Problem 3. Calculate the probability mass function of X whose cumulative distribution function is given by : 0 b < 0 / 0 b < 3/5 b < F X (b = 4/5 b < 3 9/0 3 b < 3.5 b 3.5 By definition, for any real number x, F X (x = P(X x. Here, we notice that the distribution function F X is piecewise constant, which means that the underlying random variable X is discrete and the values that X takes on correspond to points y where F X is discontinuous, that is X(Ω = {0,,, 3, 3.5}. The probability mass function p X of X can be calculated in the following fashion : For any 0 x <, / = F X (x = y x p X(x = p X (0. So p X (0 = /. For any x <, 3/5 = F X (x = y x p X(x = p X (0 + p X (. So p X ( = 3/5 p X (0 = /0. For any x < 3, 4/5 = F X (x = y x p X(x = p X (0+p X (+p X (. So p X ( = 4/5 (p X (0 + p X ( = /5. For any 3 x < 3.5, 9/0 = F X (x = y x p X(x = p X (0 + p X ( + p X ( + p X (3. So p X (3 = 9/0 (p X (0 + p X ( + p X ( = /0. Since p X is a pmf, p X (0 + p X ( + p X ( + p X (3 + p X (3.5 =. So p X (3.5 = (p X (0 + p X ( + p X ( + p X (3 = /0. Problem 4. Two cards are chosen randomly from a standard deck of fiftytwo cards. Suppose that we win $ for each spade selected and we lose $ for each heart or diamond selected, we neither win nor lose if we pick a club. Let X denote our winnings. What are the possible values of X, and what is the probability mass function of X? If you pick cards that are either hearts or diamonds, you win X = ( dollars. This happens with probability : p X ( = P(X = = 6 ( / 5 = 5/0. If you pick one spade and the other one is either heart or diamond, you win X = dollar. This happens with probability : p X ( = ( ( 3 6 ( / 5 = 3/5. If you pick one club and the other one is either heart or diamond, you win X = dollar. This happens with probability : p X ( = ( 3 ( 6 ( / 5 = 3/5. 4
5 If you pick spades, you win X = 4 dollars. This happens with probability : p X (4 = ( 3 / ( 5 = /7. If you pick a spade and the other one is a club, you win X = dollars. This happens with probability : p X ( = ( 3 ( 3 / ( 5 = 3/0. If you pick clubs, you win X = 0 dollar. This happens with probability : p X (0 = ( 3 / ( 5 = /7. Problem 5. 0 candidates 5 men and 5 women take an exam and are ranked according to their scores. Assume that no two scores are alike and all 0! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman. (For instance, X = if the top-ranked person is female. Find the probability mass function of X. For instance, consider that X = 4, that is that the highest ranking woman has rank 4, this implies that the 4 remaining women can have 6 possible ranks between 5 and 0. In that case, there are thus ( 6 4 ways the highest ranking woman has rank 4. We can generalize this finding and for a given i =,,..., 0, if the highest ranking woman has rank i, then there are ( 0 i 4 possible ranks for the 4 other women. Therefore, the corresponding probability is : P(X = i = ( 0 i 4 ( 0 5 Problem 6. We toss a coin n times (n. We denote by X the difference between the number of heads and the number of tails obtained. (a What are the possible values of X? For a given integer n, let H be the number of heads obtained when a coin is tossed n times. Then, T the number of tails obtained is T = n H. Therefore, the difference between the number of heads and the number of tails obtained is X = H T = H (n H = n + H. Since H can takes any values on {0,,..., n}, the possible values of X are { n, n +, n + 4,..., n 4, n, n}, that is one in every two integers between n and n, that is also all integers between n and n that are odd if n is odd, or all integers between n and n that are even if n is even. (b For n = 3, if the coin is assumed fair, give the probability mass function of X. If n = 3, X can take values 3,, and 3. X = 3 means that the coin lands 3 times on tails, which happens with probability : p X ( 3 = P(X = 3 = P({T T T } = (/ 3 = /8 X = means that the coin lands once on heads and twice on tails, which happens with probability : p X ( = P({HT T }+P({T HT }+ P({T T H} = 3 (/ 3 = 3/8 5
6 X = means that the coin lands twice on heads and once on tails, which happens with probability : p X ( = P({HHT } + P({HT H} + P({T HH} = 3 (/ 3 = 3/8 X = 3 means that the coin lands 3 times on heads, which happens with probability : p X (3 = P({HHH} = (/ 3 = /8 Problem 7. In a store, two customers ask a salesman advice about laptops. The first customer will purchase a laptop with probability 0.3, and independently of the first customer, the second will buy a laptop with probability 0.6. Any sale made is equally likely to be either for model A, which costs $000, or model B, which costs $500. Determine the probability mass function of X, the total dollar value of all sales. Let us define the following events : E: The salesman makes no sale, this happens with probability : P(E = ( 0.3( 0.6 = 0.8 F : The salesman makes exactly one sale, this happens with probability : P(F = 0.3 ( ( = 0.54 G: The salesman makes exactly two sales, this happens with probability : P(G = = 0.8 In addition, we denote A the event that a sale is made for model A and B the event that a sale is made for model B. We can notice that X can take 5 different values : 0, 500, 000, 500, 000 : {X = 0} is the event of $0 sales, which corresponds exactly to event E : p X (0 = P(E = 0.8 {X = 500} occurs when one sale of model B is made : p X (500 = P(F B = P(F P(B F = 0.54 / = 0.7 {X = 000} occurs when either one sale of model A is made or models B are sold: p X (000 = P(F A + P(G {BB} = P(F P(A F + P(GP({BB} G = 0.54 / (/ = 0.35 {X = 500} occurs when model A and model B are sold : p X (500 = P(G {AB, BA} = P(GP({AB, BA} G = 0.8 (/ = 0.09 {X = 000} occurs when two models A are sold: p X (000 = P(G {AA} = P(GP({AA} G = 0.8 (/ =
Math/Stat 394 Homework 5
Math/Stat 394 Homework 5 1. If we select two black balls then X 4. This happens with probability ( 4 2). If we select two white balls then X 2. This happens with ( 14 probability (8 2). If we select two
More informationChapter 4 : Discrete Random Variables
STAT/MATH 394 A - PROBABILITY I UW Autumn Quarter 2015 Néhémy Lim Chapter 4 : Discrete Random Variables 1 Random variables Objectives of this section. To learn the formal definition of a random variable.
More informationOutline Conditional Probability The Law of Total Probability and Bayes Theorem Independent Events. Week 4 Classical Probability, Part II
Week 4 Classical Probability, Part II Week 4 Objectives This week we continue covering topics from classical probability. The notion of conditional probability is presented first. Important results/tools
More informationChapter 3 : Conditional Probability and Independence
STAT/MATH 394 A - PROBABILITY I UW Autumn Quarter 2016 Néhémy Lim Chapter 3 : Conditional Probability and Independence 1 Conditional Probabilities How should we modify the probability of an event when
More information(a) Fill in the missing probabilities in the table. (b) Calculate P(F G). (c) Calculate P(E c ). (d) Is this a uniform sample space?
Math 166 Exam 1 Review Sections L.1-L.2, 1.1-1.7 Note: This review is more heavily weighted on the new material this week: Sections 1.5-1.7. For more practice problems on previous material, take a look
More informationI - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability
What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)
More informationProbability Year 10. Terminology
Probability Year 10 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationLecture Lecture 5
Lecture 4 --- Lecture 5 A. Basic Concepts (4.1-4.2) 1. Experiment: A process of observing a phenomenon that has variation in its outcome. Examples: (E1). Rolling a die, (E2). Drawing a card form a shuffled
More informationName: Exam 2 Solutions. March 13, 2017
Department of Mathematics University of Notre Dame Math 00 Finite Math Spring 07 Name: Instructors: Conant/Galvin Exam Solutions March, 07 This exam is in two parts on pages and contains problems worth
More informationProbability Year 9. Terminology
Probability Year 9 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationSTAT 516: Basic Probability and its Applications
Lecture 3: Conditional Probability and Independence Prof. Michael September 29, 2015 Motivating Example Experiment ξ consists of rolling a fair die twice; A = { the first roll is 6 } amd B = { the sum
More informationIf S = {O 1, O 2,, O n }, where O i is the i th elementary outcome, and p i is the probability of the i th elementary outcome, then
1.1 Probabilities Def n: A random experiment is a process that, when performed, results in one and only one of many observations (or outcomes). The sample space S is the set of all elementary outcomes
More informationWhat is Probability? Probability. Sample Spaces and Events. Simple Event
What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5
More information4/17/2012. NE ( ) # of ways an event can happen NS ( ) # of events in the sample space
I. Vocabulary: A. Outcomes: the things that can happen in a probability experiment B. Sample Space (S): all possible outcomes C. Event (E): one outcome D. Probability of an Event (P(E)): the likelihood
More informationProblem # Number of points 1 /20 2 /20 3 /20 4 /20 5 /20 6 /20 7 /20 8 /20 Total /150
Name Student ID # Instructor: SOLUTION Sergey Kirshner STAT 516 Fall 09 Practice Midterm #1 January 31, 2010 You are not allowed to use books or notes. Non-programmable non-graphic calculators are permitted.
More informationMutually Exclusive Events
172 CHAPTER 3 PROBABILITY TOPICS c. QS, 7D, 6D, KS Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes
More informationSTAT 414: Introduction to Probability Theory
STAT 414: Introduction to Probability Theory Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical Exercises
More informationQuick review on Discrete Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 2017 Néhémy Lim Quick review on Discrete Random Variables Notations. Z = {..., 2, 1, 0, 1, 2,...}, set of all integers; N = {0, 1, 2,...}, set of natural
More informationSTAT 418: Probability and Stochastic Processes
STAT 418: Probability and Stochastic Processes Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical
More informationArkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan
2.4 Random Variables Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan By definition, a random variable X is a function with domain the sample space and range a subset of the
More informationThe probability of an event is viewed as a numerical measure of the chance that the event will occur.
Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that
More informationCHAPTER 3 PROBABILITY: EVENTS AND PROBABILITIES
CHAPTER 3 PROBABILITY: EVENTS AND PROBABILITIES PROBABILITY: A probability is a number between 0 and 1, inclusive, that states the long-run relative frequency, likelihood, or chance that an outcome will
More informationIntroduction and basic definitions
Chapter 1 Introduction and basic definitions 1.1 Sample space, events, elementary probability Exercise 1.1 Prove that P( ) = 0. Solution of Exercise 1.1 : Events S (where S is the sample space) and are
More information4. Suppose that we roll two die and let X be equal to the maximum of the two rolls. Find P (X {1, 3, 5}) and draw the PMF for X.
Math 10B with Professor Stankova Worksheet, Midterm #2; Wednesday, 3/21/2018 GSI name: Roy Zhao 1 Problems 1.1 Bayes Theorem 1. Suppose a test is 99% accurate and 1% of people have a disease. What is the
More informationCourse: ESO-209 Home Work: 1 Instructor: Debasis Kundu
Home Work: 1 1. Describe the sample space when a coin is tossed (a) once, (b) three times, (c) n times, (d) an infinite number of times. 2. A coin is tossed until for the first time the same result appear
More information4.2 Probability Models
4.2 Probability Models Ulrich Hoensch Tuesday, February 19, 2013 Sample Spaces Examples 1. When tossing a coin, the sample space is S = {H, T }, where H = heads, T = tails. 2. When randomly selecting a
More informationLet us think of the situation as having a 50 sided fair die; any one number is equally likely to appear.
Probability_Homework Answers. Let the sample space consist of the integers through. {, 2, 3,, }. Consider the following events from that Sample Space. Event A: {a number is a multiple of 5 5, 0, 5,, }
More informationIntermediate Math Circles November 8, 2017 Probability II
Intersection of Events and Independence Consider two groups of pairs of events Intermediate Math Circles November 8, 017 Probability II Group 1 (Dependent Events) A = {a sales associate has training} B
More informationBasic Concepts of Probability
Probability Probability theory is the branch of math that deals with unpredictable or random events Probability is used to describe how likely a particular outcome is in a random event the probability
More informationHW2 Solutions, for MATH441, STAT461, STAT561, due September 9th
HW2 Solutions, for MATH44, STAT46, STAT56, due September 9th. You flip a coin until you get tails. Describe the sample space. How many points are in the sample space? The sample space consists of sequences
More information11. Probability Sample Spaces and Probability
11. Probability 11.1 Sample Spaces and Probability 1 Objectives A. Find the probability of an event. B. Find the empirical probability of an event. 2 Theoretical Probabilities 3 Example A fair coin is
More information5. Conditional Distributions
1 of 12 7/16/2009 5:36 AM Virtual Laboratories > 3. Distributions > 1 2 3 4 5 6 7 8 5. Conditional Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an
More informationExam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3)
1 Exam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3) On this exam, questions may come from any of the following topic areas: - Union and intersection of sets - Complement of
More informationYear 10 Mathematics Probability Practice Test 1
Year 10 Mathematics Probability Practice Test 1 1 A letter is chosen randomly from the word TELEVISION. a How many letters are there in the word TELEVISION? b Find the probability that the letter is: i
More informationM378K In-Class Assignment #1
The following problems are a review of M6K. M7K In-Class Assignment # Problem.. Complete the definition of mutual exclusivity of events below: Events A, B Ω are said to be mutually exclusive if A B =.
More informationConditional Probability & Independence. Conditional Probabilities
Conditional Probability & Independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Definition: the conditional probability of E given F is P(E F
More informationChapter. Probability
Chapter 3 Probability Section 3.1 Basic Concepts of Probability Section 3.1 Objectives Identify the sample space of a probability experiment Identify simple events Use the Fundamental Counting Principle
More informationUNIT 5 ~ Probability: What Are the Chances? 1
UNIT 5 ~ Probability: What Are the Chances? 1 6.1: Simulation Simulation: The of chance behavior, based on a that accurately reflects the phenomenon under consideration. (ex 1) Suppose we are interested
More informationReview Basic Probability Concept
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.9008 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationMotivation. Stat Camp for the MBA Program. Probability. Experiments and Outcomes. Daniel Solow 5/10/2017
Stat Camp for the MBA Program Daniel Solow Lecture 2 Probability Motivation You often need to make decisions under uncertainty, that is, facing an unknown future. Examples: How many computers should I
More informationMonty Hall Puzzle. Draw a tree diagram of possible choices (a possibility tree ) One for each strategy switch or no-switch
Monty Hall Puzzle Example: You are asked to select one of the three doors to open. There is a large prize behind one of the doors and if you select that door, you win the prize. After you select a door,
More informationIM3 DEC EXAM PREP MATERIAL DEC 2016
1. Given the line x 3 y 5 =1; Paper 1 - CALCULATOR INACTIVE a. Determine the slope of this line. b. Write the equation of this line in function form. c. Evaluate f ( 12). d. Solve for x if 95 = f (x).
More informationStatistics 100 Exam 2 March 8, 2017
STAT 100 EXAM 2 Spring 2017 (This page is worth 1 point. Graded on writing your name and net id clearly and circling section.) PRINT NAME (Last name) (First name) net ID CIRCLE SECTION please! L1 (MWF
More informationMAT 271E Probability and Statistics
MAT 271E Probability and Statistics Spring 2011 Instructor : Class Meets : Office Hours : Textbook : Supp. Text : İlker Bayram EEB 1103 ibayram@itu.edu.tr 13.30 16.30, Wednesday EEB? 10.00 12.00, Wednesday
More information2. AXIOMATIC PROBABILITY
IA Probability Lent Term 2. AXIOMATIC PROBABILITY 2. The axioms The formulation for classical probability in which all outcomes or points in the sample space are equally likely is too restrictive to develop
More informationSTA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru Venkateswara Rao, Ph.D. STA 2023 Fall 2016 Venkat Mu ALL THE CONTENT IN THESE SOLUTIONS PRESENTED IN BLUE AND BLACK
More informationELEG 3143 Probability & Stochastic Process Ch. 1 Probability
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 1 Probability Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Applications Elementary Set Theory Random
More informationIEOR 3106: Introduction to Operations Research: Stochastic Models. Professor Whitt. SOLUTIONS to Homework Assignment 1
IEOR 3106: Introduction to Operations Research: Stochastic Models Professor Whitt SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability
More informationIndependence. P(A) = P(B) = 3 6 = 1 2, and P(C) = 4 6 = 2 3.
Example: A fair die is tossed and we want to guess the outcome. The outcomes will be 1, 2, 3, 4, 5, 6 with equal probability 1 6 each. If we are interested in getting the following results: A = {1, 3,
More information(i) Given that a student is female, what is the probability of having a GPA of at least 3.0?
MATH 382 Conditional Probability Dr. Neal, WKU We now shall consider probabilities of events that are restricted within a subset that is smaller than the entire sample space Ω. For example, let Ω be the
More informationProbability & Random Variables
& Random Variables Probability Probability theory is the branch of math that deals with random events, processes, and variables What does randomness mean to you? How would you define probability in your
More informationComputations - Show all your work. (30 pts)
Math 1012 Final Name: Computations - Show all your work. (30 pts) 1. Fractions. a. 1 7 + 1 5 b. 12 5 5 9 c. 6 8 2 16 d. 1 6 + 2 5 + 3 4 2.a Powers of ten. i. 10 3 10 2 ii. 10 2 10 6 iii. 10 0 iv. (10 5
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 3 Probability Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive
More informationStat 225 Week 1, 8/20/12-8/24/12, Notes: Set Theory
Stat 225 Week 1, 8/20/12-8/24/12, Notes: Set Theory The Fall 2012 Stat 225 T.A.s September 7, 2012 The material in this handout is intended to cover general set theory topics. Information includes (but
More informationName: Firas Rassoul-Agha
Midterm 1 - Math 5010 - Spring 016 Name: Firas Rassoul-Agha Solve the following 4 problems. You have to clearly explain your solution. The answer carries no points. Only the work does. CALCULATORS ARE
More informationConditional probability
CHAPTER 4 Conditional probability 4.1. Introduction Suppose there are 200 men, of which 100 are smokers, and 100 women, of which 20 are smokers. What is the probability that a person chosen at random will
More informationChapter 2 Random Variables
Stochastic Processes Chapter 2 Random Variables Prof. Jernan Juang Dept. of Engineering Science National Cheng Kung University Prof. Chun-Hung Liu Dept. of Electrical and Computer Eng. National Chiao Tung
More informationDetermining Probabilities. Product Rule for Ordered Pairs/k-Tuples:
Determining Probabilities Product Rule for Ordered Pairs/k-Tuples: Determining Probabilities Product Rule for Ordered Pairs/k-Tuples: Proposition If the first element of object of an ordered pair can be
More informationLecture 3 Probability Basics
Lecture 3 Probability Basics Thais Paiva STA 111 - Summer 2013 Term II July 3, 2013 Lecture Plan 1 Definitions of probability 2 Rules of probability 3 Conditional probability What is Probability? Probability
More informationSTEP Support Programme. Statistics STEP Questions: Solutions
STEP Support Programme Statistics STEP Questions: Solutions 200 S Q2 Preparation (i) (a) The sum of the probabilities is, so we have k + 2k + 3k + 4k k 0. (b) P(X 3) P(X 3) + P(X 4) 7 0. (c) E(X) 0 ( +
More informationMath 151. Rumbos Fall Solutions to Review Problems for Exam 2. Pr(X = 1) = ) = Pr(X = 2) = Pr(X = 3) = p X. (k) =
Math 5. Rumbos Fall 07 Solutions to Review Problems for Exam. A bowl contains 5 chips of the same size and shape. Two chips are red and the other three are blue. Draw three chips from the bowl at random,
More informationIntroduction to Probability, Fall 2009
Introduction to Probability, Fall 2009 Math 30530 Review questions for exam 1 solutions 1. Let A, B and C be events. Some of the following statements are always true, and some are not. For those that are
More informationSTA 2023 EXAM-2 Practice Problems. Ven Mudunuru. From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru, Venkateswara Rao STA 2023 Spring 2016 1 1. A committee of 5 persons is to be formed from 6 men and 4 women. What
More informationSTAT 516 Answers Homework 2 January 23, 2008 Solutions by Mark Daniel Ward PROBLEMS. = {(a 1, a 2,...) : a i < 6 for all i}
STAT 56 Answers Homework 2 January 23, 2008 Solutions by Mark Daniel Ward PROBLEMS 2. We note that E n consists of rolls that end in 6, namely, experiments of the form (a, a 2,...,a n, 6 for n and a i
More informationAnnouncements. Lecture 5: Probability. Dangling threads from last week: Mean vs. median. Dangling threads from last week: Sampling bias
Recap Announcements Lecture 5: Statistics 101 Mine Çetinkaya-Rundel September 13, 2011 HW1 due TA hours Thursday - Sunday 4pm - 9pm at Old Chem 211A If you added the class last week please make sure to
More informationReview of Probability. CS1538: Introduction to Simulations
Review of Probability CS1538: Introduction to Simulations Probability and Statistics in Simulation Why do we need probability and statistics in simulation? Needed to validate the simulation model Needed
More informationCHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS
CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS 4.2 Events and Sample Space De nition 1. An experiment is the process by which an observation (or measurement) is obtained Examples 1. 1: Tossing a pair
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER / Probability
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER 2 2017/2018 DR. ANTHONY BROWN 5.1. Introduction to Probability. 5. Probability You are probably familiar with the elementary
More informationEXAM. Exam #1. Math 3342 Summer II, July 21, 2000 ANSWERS
EXAM Exam # Math 3342 Summer II, 2 July 2, 2 ANSWERS i pts. Problem. Consider the following data: 7, 8, 9, 2,, 7, 2, 3. Find the first quartile, the median, and the third quartile. Make a box and whisker
More information6.041/6.431 Spring 2009 Quiz 1 Wednesday, March 11, 7:30-9:30 PM. SOLUTIONS
6.0/6.3 Spring 009 Quiz Wednesday, March, 7:30-9:30 PM. SOLUTIONS Name: Recitation Instructor: Question Part Score Out of 0 all 0 a 5 b c 5 d 5 e 5 f 5 3 a b c d 5 e 5 f 5 g 5 h 5 Total 00 Write your solutions
More informationIntroduction to Probability 2017/18 Supplementary Problems
Introduction to Probability 2017/18 Supplementary Problems Problem 1: Let A and B denote two events with P(A B) 0. Show that P(A) 0 and P(B) 0. A A B implies P(A) P(A B) 0, hence P(A) 0. Similarly B A
More informationConditional Probability & Independence. Conditional Probabilities
Conditional Probability & Independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Definition: the conditional probability of E given F is P(E F
More informationProbability Theory Review
Cogsci 118A: Natural Computation I Lecture 2 (01/07/10) Lecturer: Angela Yu Probability Theory Review Scribe: Joseph Schilz Lecture Summary 1. Set theory: terms and operators In this section, we provide
More informationStatistics for Managers Using Microsoft Excel (3 rd Edition)
Statistics for Managers Using Microsoft Excel (3 rd Edition) Chapter 4 Basic Probability and Discrete Probability Distributions 2002 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic probability concepts
More informationProbability Distributions for Discrete RV
An example: Assume we toss a coin 3 times and record the outcomes. Let X i be a random variable defined by { 1, if the i th outcome is Head; X i = 0, if the i th outcome is Tail; Let X be the random variable
More informationLecture 1 Introduction to Probability and Set Theory Text: A Course in Probability by Weiss
Lecture 1 to and Set Theory Text: A Course in by Weiss 1.2 2.3 STAT 225 to Models January 13, 2014 to and Whitney Huang Purdue University 1.1 Agenda to and 1 2 3 1.2 Motivation Uncertainty/Randomness in
More informationDiscrete Distributions
Discrete Distributions STA 281 Fall 2011 1 Introduction Previously we defined a random variable to be an experiment with numerical outcomes. Often different random variables are related in that they have
More informationDiscrete Probability
Discrete Probability Counting Permutations Combinations r- Combinations r- Combinations with repetition Allowed Pascal s Formula Binomial Theorem Conditional Probability Baye s Formula Independent Events
More informationChapter 2.5 Random Variables and Probability The Modern View (cont.)
Chapter 2.5 Random Variables and Probability The Modern View (cont.) I. Statistical Independence A crucially important idea in probability and statistics is the concept of statistical independence. Suppose
More information3.2 Probability Rules
3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need
More informationCh 14 Randomness and Probability
Ch 14 Randomness and Probability We ll begin a new part: randomness and probability. This part contain 4 chapters: 14-17. Why we need to learn this part? Probability is not a portion of statistics. Instead
More informationTOPIC 12 PROBABILITY SCHEMATIC DIAGRAM
TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos
More informationProbability and Statistics Notes
Probability and Statistics Notes Chapter One Jesse Crawford Department of Mathematics Tarleton State University (Tarleton State University) Chapter One Notes 1 / 71 Outline 1 A Sketch of Probability and
More informationProblems and results for the ninth week Mathematics A3 for Civil Engineering students
Problems and results for the ninth week Mathematics A3 for Civil Engineering students. Production line I of a factor works 0% of time, while production line II works 70% of time, independentl of each other.
More informationChapter 5, 6 and 7: Review Questions: STAT/MATH Consider the experiment of rolling a fair die twice. Find the indicated probabilities.
Chapter5 Chapter 5, 6 and 7: Review Questions: STAT/MATH3379 1. Consider the experiment of rolling a fair die twice. Find the indicated probabilities. (a) One of the dice is a 4. (b) Sum of the dice equals
More informationEECS 126 Probability and Random Processes University of California, Berkeley: Spring 2015 Abhay Parekh February 17, 2015.
EECS 126 Probability and Random Processes University of California, Berkeley: Spring 2015 Abhay Parekh February 17, 2015 Midterm Exam Last name First name SID Rules. You have 80 mins (5:10pm - 6:30pm)
More informationExam 1 - Math Solutions
Exam 1 - Math 3200 - Solutions Spring 2013 1. Without actually expanding, find the coefficient of x y 2 z 3 in the expansion of (2x y z) 6. (A) 120 (B) 60 (C) 30 (D) 20 (E) 10 (F) 10 (G) 20 (H) 30 (I)
More informationDeep Learning for Computer Vision
Deep Learning for Computer Vision Lecture 3: Probability, Bayes Theorem, and Bayes Classification Peter Belhumeur Computer Science Columbia University Probability Should you play this game? Game: A fair
More informationLecture 2: Probability. Readings: Sections Statistical Inference: drawing conclusions about the population based on a sample
Lecture 2: Probability Readings: Sections 5.1-5.3 1 Introduction Statistical Inference: drawing conclusions about the population based on a sample Parameter: a number that describes the population a fixed
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Lecture 2 Probability Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 3-1 3.1 Definition Random Experiment a process leading to an uncertain
More information1 Presessional Probability
1 Presessional Probability Probability theory is essential for the development of mathematical models in finance, because of the randomness nature of price fluctuations in the markets. This presessional
More informationMA 250 Probability and Statistics. Nazar Khan PUCIT Lecture 15
MA 250 Probability and Statistics Nazar Khan PUCIT Lecture 15 RANDOM VARIABLES Random Variables Random variables come in 2 types 1. Discrete set of outputs is real valued, countable set 2. Continuous set
More informationMath , Fall 2012: HW 5 Solutions
Math 230.0, Fall 202: HW 5 Solutions Due Thursday, October 4th, 202. Problem (p.58 #2). Let X and Y be the numbers obtained in two draws at random from a box containing four tickets labeled, 2, 3, 4. Display
More informationOutline. Probability. Math 143. Department of Mathematics and Statistics Calvin College. Spring 2010
Outline Math 143 Department of Mathematics and Statistics Calvin College Spring 2010 Outline Outline 1 Review Basics Random Variables Mean, Variance and Standard Deviation of Random Variables 2 More Review
More informationTopic 5: Probability. 5.4 Combined Events and Conditional Probability Paper 1
Topic 5: Probability Standard Level 5.4 Combined Events and Conditional Probability Paper 1 1. In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music. The Venn
More informationPROBABILITY.
PROBABILITY PROBABILITY(Basic Terminology) Random Experiment: If in each trial of an experiment conducted under identical conditions, the outcome is not unique, but may be any one of the possible outcomes,
More informationChapter 01: Probability Theory (Cont d)
Chapter 01: Probability Theory (Cont d) Section 1.5: Probabilities of Event Intersections Problem (01): When a company receives an order, there is a probability of 0.42 that its value is over $1000. If
More informationLecture 3. Probability and elements of combinatorics
Introduction to theory of probability and statistics Lecture 3. Probability and elements of combinatorics prof. dr hab.inż. Katarzyna Zakrzewska Katedra Elektroniki, AGH e-mail: zak@agh.edu.pl http://home.agh.edu.pl/~zak
More informationBasic Probability. Introduction
Basic Probability Introduction The world is an uncertain place. Making predictions about something as seemingly mundane as tomorrow s weather, for example, is actually quite a difficult task. Even with
More informationDiscrete Random Variables. Discrete Random Variables
Random Variables In many situations, we are interested in numbers associated with the outcomes of a random experiment. For example: Testing cars from a production line, we are interested in variables such
More information