MATH 201. Final ANSWERS August 12, 2016


 Estella Poole
 1 years ago
 Views:
Transcription
1 MATH 01 Final ANSWERS August 1, 016 Part A points) A bag contains three different types of dice: four 6sided dice, five 8sided dice, and six 0sided dice. A die is drawn from the bag and then rolled. Assume that each die is equally likely to be drawn from the bag. a) What is the probability of rolling a? b) If the outcome of the roll is a 5, what is the probability that an 8sided die was rolled? a) Let X denote the outcome of the roll. Let E 1 be the event of rolling a sixsided die, E the event of rolling an 8sided die, and E 3 the event of rolling a 0sided die. We are given that P E 1 ) = 4/15, P E ) = 5/15 and P E 3 ) = 6/15. By writing Ω = E 1 E E 3, we find P X = ) = P X = E 1 )P E 1 ) + P X = E )P E ) + P X = E 3 )P E 3 ) = 04/15) + 05/15) + 1/0)6/15) = 1/50. b) Using Bayes formula, we find P X = 5 E )P E ) P E X = 5) = P X = 5 E 1 )P E 1 ) + P X = 5 E )P E ) + P X = 5 E 3 )P E 3 ) 1/8)5/15) = 1/6)4/15) + 1/8)5/15) + 1/0)6/15) = points) Suppose X is a random variable such that EX = 4 and VarX = 3. Compute E3X ) ). 1
2 We compute E3X ) ) = E9X 1X + 4) = 9EX 1EX + 4 = 9VarX) + EX) ) 1EX + 4 = ) 14) + 4 = points) The floor function x = max{n Z : n x} gives the largest integer less than or equal to x. For example 3.5 = 3 and 4 = 4. Suppose X is a continuous random variable with probability density function given by 1 if x 1, x fx) = 0 if x < 1. a) Find the probability mass function p X k) for X. b) Is E X <? Justify your answer. a) Since X 1 with probability 1, we clearly have p X k) = P X = k) = 0 for all integer k 0. For each integer k 1 we find p X k) = P X = k) = P X [k, k + 1)) = = 1 kk + 1). k+1 k [ dx x = 1 ] k+1 = 1 x k k 1 k + 1 b) No. Using part a) we compute E X = kp X = k) = k=1 1 k kk + 1) = 1 k + 1, k=1 k=1 and this is the divergent harmonic series, indicating that E X = points) Suppose you play cards with a standard 5card deck. In this game, 4 players are each dealt 13 cards, with all outcomes equally likely. A standard deck of cards contains 4 suits: clubs, diamonds, hearts, and spades. Each suit has the cards through 10, and a jack, queen, king, and an ace.
3 a) What is the chance that you get all 4 aces and all 4 kings? b) What is the chance of getting aces and kings? a) It seems best to set up the probability space using sampling without replacement, order doesn t matter. There are 5 choose 13 ways to get a 13card hand. There is only one way to get 4 aces and 4 kings, and 58) choose 5 ways of getting the 5 remaining cards. Therefore, the probability is 44 ) 5 ) 5 13 b) We set up the probability space the same way as in part a). There are 4 choose ways of getting the aces, and 4 choose ways of getting the kings. There are 54) choose 9 ways of getting the remaining 9 cards. So the probability is 4 ) 44 ) 9 5 ) points) Consider the circuit drawn below. 1 A 4 3 B Electricity flows from A to B if there is at least one path available. Let p i be the probability that switch i is closed, so electricity can flow through it. Assume that the switches are open or closed independently of each other. In terms of p 1,..., p 4, what is the probability that electricity can flow from A to B? Hint: You may want to use the inclusionexclusion formula. Let A i be the event that switch i is closed. Let F be the event that electricity can flow 3
4 through the circuit. Since there are 3 possible paths for the electricity, we have F = A 1 A ) A 1 A 3 ) A 4 Using the inclusionexclusion formula and the independence of switches, we get P F ) =P A 1 A ) + P A 1 A 3 ) + P A 4 ) P A 1 A A 3 ) P A 1 A A 4 ) P A 1 A 3 A 4 ) + P A 1 A A 3 A 4 ) =p 1 p + p 1 p 3 + p 4 p 1 p p 3 p 1 p p 4 p 1 p 3 p 4 + p 1 p p 3 p points) Suppose that you are snowed in on a Saturday night at the dorms, and to kill time you flip coins. Usually they land on heads or tails, but on rare occasions they land on their edge. The probability of getting an edge on a given flip is 1/100, and these events are independent. Let A be the event that you get 55 edges in 5000 flips. a) Write an expression for the exact probability of A. b) Use the Poisson approximation to approximate the probability of A. a) Using the binomial probabilities, 5000 P A) = 55 ) ) 55 ) b) Using the Poisson approximation for the binomial, we find that λ = np = 5000/100 = 50, so Part B P A) ! e points) Suppose that the joint probability mass function p X,Y i, j) = pi, j) for random variables X, Y is given as follows. p1, 1) = 0.1 p1, ) = 0. p1, 3) = 0.3 p, 1) = 0.1 p, ) = 0. p, 3) = 0.1 a) Find EX. 4
5 b) Find EY. c) Find CovX, Y ) a) According to the above table, X only takes on the values 1 and. Summing the probabilities in the first row of the table, we find that P X = 1) = 0.6. Summing the second row, we find P X = ) = 0.4. So EX = = 1.4 b) Summing the probabilities in the columns of the table, we find that P Y = 1) = 0., P Y = ) = 0.4, P Y = 3) = 0.4. Thus, EY = =. c) Recall that CovX, Y ) = E[XY ] EX)EY ). First we compute E[XY ] using the table. E[XY ] = = 3 Therefore CovX, Y ) = E[XY ] EX)EY ) = = points) A store opens at 8:00am. Assume that customers arrive according to a Poisson process with parameter λ, with time measured in minutes. Suppose that the last customer to arrive before 9:00am arrives at T minutes before 9:00am. If no customers arrived between 8:00am and 9:00am, let T = 60 1 hour). Find the cumulative distribution function cdf) of T. Hint: Recall that P T > t) = 1 F T t). First, it is clear that T 0 with probability 1, so F T t) = 0 if t < 0. So assume that t 0. Also, when t > 60, since T is at most 60, we get F T t) = P T t) = 1. Since customer arrivals are modeled by a Poisson process NI), we see that if 0 t 60 then P T > t) = P N[60 t, 60]) = 0) = e λt. 5
6 So, if 0 t 60 then F T t) = 1 P T > t) = 1 e λt. To summarize, 0 if t < 0 F T t) = 1 e λt if 0 t 60 1 if t > points) Suppose that X 1, X,... are i.i.d. independent and identically distributed) with mean µ = 3 and variance σ = 5. Recall that X = X 1+ +X n n µ. For n = 95, approximate P X 3.5) is a common estimator for Express your answer in terms of Φx), the cumulative distribution function of the standard normal distribution for positive values of x. We let S n = X X n, and recall that the central limit theorem says that for large values of n, S n nµ σ n is approximately distributed according to the standard normal distribution. Therefore, P X 3.5) = P S ) n n 3.5 We subtract µ = 3 from all 3 terms on the right, divide by σ = 5, and finally multiply by n = 95. This gives us P S ) n n 3.5 = P 3 S n nµ n 3 = P = P 5 S n nµ σn ) ) 5 3) 95 S n nµ 5 σ n = P S ) n nµ σ n P ) Z ) 3.5 3)
7 where Z is a N0, 1) random variable. Expressing the probability in terms of Φ, we get P S ) n n 3.5 P ) Z ) = Φ Φ ) ) [ )] = Φ 1 Φ ) ) = Φ + Φ points) Consider the triangle D with vertices 0, 0), 1, 0) and 1, 1). Suppose X, Y ) is a random point chosen uniformly from the triangle D. a) Find the marginal densities of X and Y. b) Are X and Y independent? Justify your answer. y 1, 1) D 0, 0) 1, 0) x a) Since the triangle has area 1/, the joint probability density function of X, Y ) is given by if x, y) D, f X,Y x, y) = 0 if x, y) / D. 7
8 We compute the marginal densities through integration x dy if x [0, 1] 0 f X x) = f X,Y x, y)dy = 0 if x / [0, 1] x if x [0, 1] = 0 if x / [0, 1]. 1 dy if y [0, 1] y f Y y) = f X,Y x, y)dx = 0 if y / [0, 1] 1 y) if y [0, 1] = 0 if y / [0, 1]. b) Note that f X,Y x, y) f X x)f Y y) for some x, y) D, and it follows that X and Y are not independent points) Suppose that X and Y are independent random variables. Assume that X has range {1,, 3} with probability mass function P X = 1) = 1 4 P X = ) = 1 P X = 3) = 1 4, and assume that Y has range {, 3, 4} with probability mass function P Y = ) = 1 P Y = 3) = 1 6 P Y = 4) = 1 3. a) Find the moment generating functions M X t) and M Y t) of X and Y, respectively. b) Compute M X+Y t) and use this to identify the range and probability mass function of X + Y. a) We compute M X t) = Ee tx ) = e t P X = 1) + e t P X = ) + e 3t P X = 3) = 1 4 et + 1 et e3t, 8
9 and M Y t) = Ee ty ) = e t P Y = ) + e 3t P Y = 3) + e 4t P Y = 4) = 1 et e3t e4t. b) Since X and Y are independent, we find 1 M X+Y t) = M X t)m Y t) = 4 et + 1 et + 1 ) 1 4 e3t et e3t + 1 ) 3 e4t = 1 8 e3t e4t e5t e6t e7t. It follows that X + Y has range {3, 4, 5, 6, 7} with the probability mass function P X + Y = 3) = 1 8 P X + Y = 6) = 5 4 P X + Y = 4) = P X + Y = 5) = 7 4 P X + Y = 7) = points) Suppose X Poisson6) and Y Exp1/6) with CovX, Y ) = 1. a) Use Markov s inequality to obtain an upper bound on P X + Y > 0). b) Find the variance VarX + Y ). c) Use Chebyshev s inequality to obtain another upper bound on P X + Y > 0). a) Recall that if X Poissonλ), then EX = λ and VarX = λ. In this case EX = VarX = 6. Also, if Y Expλ), then EX = 1/λ and VarX = 1/λ. In this case EY = 6 and VarY = 36. Note that X 0 and Y 0, so we also have X + Y 0. By Markov s inequality, we find P X + Y > 0) EX + Y ) 0 = EX + EY 0 = 1 0 = 3 5. b) We compute VarX + Y ) = CovX + Y, X + Y ) = VarX) + CovX, Y ) + VarY ) = 6 + 1) + 36 = 18. 9
10 c) By Chebyshev s inequality, since EX + Y ) = 1, we find P X + Y > 0) = P X + Y > 1 + 8) P X + Y 1 > 8) VarX + Y ) 8 = =
Joint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 101634501 Probability and Statistics for Engineers Winter 20102011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete
More informationExamination 110 Probability and Statistics Examination
Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiplechoice test questions. The test is a threehour examination
More informationMath 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
More informationFor a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )
Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (19031987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll
More information3 Multiple Discrete Random Variables
3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f
More informationChapters 5. Multivariate Probability Distributions
Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are
More informationJoint Exam 1/P Sample Exam 1
Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question
More information6. Jointly Distributed Random Variables
6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.
More informationLesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationRemember to leave your answers as unreduced fractions.
Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,
More informationBivariate Distributions
Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is
More informationDISCRETE RANDOM VARIABLES
DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced
More informationST 371 (VIII): Theory of Joint Distributions
ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or
More informationTopic 4: Multivariate random variables. Multiple random variables
Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly
More informationRandom 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
More informationSection 6.1 Joint Distribution Functions
Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function
More informationJointly Distributed Random Variables
Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................
More informationLecture Notes 1. Brief Review of Basic Probability
Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters 3 are a review. I will assume you have read and understood Chapters 3. Here is a very
More informationJoint distributions Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014
Joint distributions Math 17 Probability and Statistics Prof. D. Joyce, Fall 14 Today we ll look at joint random variables and joint distributions in detail. Product distributions. If Ω 1 and Ω are sample
More informationP (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )
Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =
More informationRANDOM VARIABLES MATH CIRCLE (ADVANCED) 3/3/2013. 3 k) ( 52 3 )
RANDOM VARIABLES MATH CIRCLE (ADVANCED) //0 0) a) Suppose you flip a fair coin times. i) What is the probability you get 0 heads? ii) head? iii) heads? iv) heads? For = 0,,,, P ( Heads) = ( ) b) Suppose
More informationProbability 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 informationStatistiek (WISB361)
Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17
More informationMTH135/STA104: Probability
MTH135/STA14: Probability Homework # 8 Due: Tuesday, Nov 8, 5 Prof Robert Wolpert 1 Define a function f(x, y) on the plane R by { 1/x < y < x < 1 f(x, y) = other x, y a) Show that f(x, y) is a joint probability
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationHoover High School Math League. Counting and Probability
Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches
More informationContemporary Mathematics MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More information( ) = P Z > = P( Z > 1) = 1 Φ(1) = 1 0.8413 = 0.1587 P X > 17
4.6 I company that manufactures and bottles of apple juice uses a machine that automatically fills 6 ounce bottles. There is some variation, however, in the amounts of liquid dispensed into the bottles
More informationLecture 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 informationStatistics 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 informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.23, 2.1 Normal distribution Math 283 / October
More informationMathematical 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 informationWorked examples Multiple Random Variables
Worked eamples Multiple Random Variables Eample Let X and Y be random variables that take on values from the set,, } (a) Find a joint probability mass assignment for which X and Y are independent, and
More informationStatistics 100A Homework 8 Solutions
Part : Chapter 7 Statistics A Homework 8 Solutions Ryan Rosario. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, the onehalf
More informationContemporary Mathematics Online Math 1030 Sample Exam I Chapters 1214 No Time Limit No Scratch Paper Calculator Allowed: Scientific
Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 1214 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the lefthand margin. You
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationm (t) = e nt m Y ( t) = e nt (pe t + q) n = (pe t e t + qe t ) n = (qe t + p) n
1. For a discrete random variable Y, prove that E[aY + b] = ae[y] + b and V(aY + b) = a 2 V(Y). Solution: E[aY + b] = E[aY] + E[b] = ae[y] + b where each step follows from a theorem on expected value from
More informationStatistics 100A Homework 7 Solutions
Chapter 6 Statistics A Homework 7 Solutions Ryan Rosario. A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 5 percent will purchase
More informationMT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...
MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More informationECE302 Spring 2006 HW7 Solutions March 11, 2006 1
ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 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 italics where
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationCovariance and Correlation. Consider the joint probability distribution f XY (x, y).
Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Section 52 Covariance and Correlation Consider the joint probability distribution f XY (x, y). Is there a relationship between X and Y? If so, what kind?
More informationFeb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172179)
Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities
More informationMath 2020 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More information6.041/6.431 Spring 2008 Quiz 2 Wednesday, April 16, 7:309:30 PM. SOLUTIONS
6.4/6.43 Spring 28 Quiz 2 Wednesday, April 6, 7:39:3 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
More informationMAS108 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
More informationMath 3C Homework 3 Solutions
Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard
More informationQuestion: What is the probability that a fivecard 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 informationExercises with solutions (1)
Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 23, Probability and Statistics, March 2015. Due:March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 3, Probability and Statistics, March 05. Due:March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More informationMULTIVARIATE 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 information4. Joint Distributions
Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose
More informationMath 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions
Math 370, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the
More informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationCommon probability distributionsi Math 217/218 Probability and Statistics Prof. D. Joyce, 2016
Introduction. ommon probability distributionsi Math 7/8 Probability and Statistics Prof. D. Joyce, 06 I summarize here some of the more common distributions used in probability and statistics. Some are
More information4. Joint Distributions of Two Random Variables
4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint
More informationWhat 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
More informationMath 2001 Homework #10 Solutions
Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal
More informationChapter 4. Multivariate Distributions
1 Chapter 4. Multivariate Distributions Joint p.m.f. (p.d.f.) Independent Random Variables Covariance and Correlation Coefficient Expectation and Covariance Matrix Multivariate (Normal) Distributions Matlab
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability
More informationStatistics  Written Examination MEC Students  BOVISA
Statistics  Written Examination MEC Students  BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.
More informationFeb 28 Homework Solutions Math 151, Winter 2012. Chapter 6 Problems (pages 287291)
Feb 8 Homework Solutions Math 5, Winter Chapter 6 Problems (pages 879) Problem 6 bin of 5 transistors is known to contain that are defective. The transistors are to be tested, one at a time, until the
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More informationLesson 5 Chapter 4: Jointly Distributed Random Variables
Lesson 5 Chapter 4: Jointly Distributed Random Variables Department of Statistics The Pennsylvania State University 1 Marginal and Conditional Probability Mass Functions The Regression Function Independence
More informationSolutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG820). December 15, 2012.
Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG8). December 15, 12. 1. (3p) The joint distribution of the discrete random variables X and
More informationMath/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
More informationProblem 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 backofchapter exercises from
More informationSTAT 430/510 Probability Lecture 14: Joint Probability Distribution, Continuous Case
STAT 430/510 Probability Lecture 14: Joint Probability Distribution, Continuous Case Pengyuan (Penelope) Wang June 20, 2011 Joint density function of continuous Random Variable When X and Y are two continuous
More informationThe 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 informationCombinatorial Probability
Chapter 2 Combinatorial Probability 2.1 Permutations and combinations As usual we begin with a question: Example 2.1. The New York State Lottery picks 6 numbers out of 54, or more precisely, a machine
More informationStatistics 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 informationJoint Distributions. Tieming Ji. Fall 2012
Joint Distributions Tieming Ji Fall 2012 1 / 33 X : univariate random variable. (X, Y ): bivariate random variable. In this chapter, we are going to study the distributions of bivariate random variables
More informationMath 425 (Fall 08) Solutions Midterm 2 November 6, 2008
Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the
More informationJoint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single
More informationPROBABILITIES AND PROBABILITY DISTRIBUTIONS
Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL
More informationLab 11. Simulations. The Concept
Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that
More informationHomework 6 (due November 4, 2009)
Homework 6 (due November 4, 2009 Problem 1. On average, how many independent games of poker are required until a preassigned player is dealt a straight? Here we define a straight to be cards of consecutive
More informationSome special discrete probability distributions
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that
More information2. Discrete random variables
2. Discrete random variables Statistics and probability: 21 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More informationPROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA
PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet
More informationMath 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 informationMATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker)
Last modified: February, 00 References: MATHEMATICS 5, SPRING 00 PROBABILITY THEORY Outline # (Combinatorics, bridge, poker) PRP(Probability and Random Processes, by Grimmett and Stirzaker), Section.7.
More informationStochastic Processes and Advanced Mathematical Finance. Laws of Large Numbers
Steven R. Dunbar Department of Mathematics 203 Avery Hall University of NebraskaLincoln Lincoln, NE 685880130 http://www.math.unl.edu Voice: 4024723731 Fax: 4024728466 Stochastic Processes and Advanced
More informationSECTION 105 Multiplication Principle, Permutations, and Combinations
105 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial
More informationBinomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution
Binomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution In probability theory and statistics, the binomial distribution is the discrete probability distribution
More informationMath 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141
Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard
More informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
More information1. Let A, B and C are three events such that P(A) = 0.45, P(B) = 0.30, P(C) = 0.35,
1. Let A, B and C are three events such that PA =.4, PB =.3, PC =.3, P A B =.6, P A C =.6, P B C =., P A B C =.7. a Compute P A B, P A C, P B C. b Compute P A B C. c Compute the probability that exactly
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
More informationBasic Probability Theory II
RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample
More informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a datagenerating
More informationProbability: 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
More informationExamples: Joint Densities and Joint Mass Functions Example 1: X and Y are jointly continuous with joint pdf
AMS 3 Joe Mitchell Eamples: Joint Densities and Joint Mass Functions Eample : X and Y are jointl continuous with joint pdf f(,) { c 2 + 3 if, 2, otherwise. (a). Find c. (b). Find P(X + Y ). (c). Find marginal
More informationLecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. LogNormal Distribution
Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Logormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch.  Problems to look at Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability
More informationStatistics 100A Homework 2 Solutions
Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6
More information