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


 Jayson McKenzie
 3 years ago
 Views:
Transcription
1 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 what the outcome will be of a physical experiment or recorded data. We find it useful to quantify the outcomes with real numbers. The function (or map or rule) that defines which real number gets associated with which particular outcome is what we call a random variable (rv) often denoted by a capital letter such as X or Y (the generic choices). Random variables are not random! The only thing that is uncertain about them is what the input will be as that comes from a yettobeperformed physical experiment or datum recorded from a notyetchosen member of a population! A random variable is NOT RANDOM! IT IS NOT RANDOM! It is a welldefined function! For example, we might say that we are interested in the heights of students in this class. I would represent the recorded height as the output of the random variable X. The only reason I am unsure of the outputs of X is that I do not know who will be chosen, but once a student is chosen, there is nothing random about this student s height. Once we have settled upon what the random variable is (i.e. how we map outcomes from a sample space, which is nothing more than a domain containing all the possible outcomes, to the real numbers), we are interested in the distribution of this random variable. Specifically, we want to know how to compute probabilities of events defined by some sets of real numbers. An event defined in terms of the random variable belonging to some set of real numbers means nothing more than the event of all outcomes in the sample space that get mapped into this set. For example, we might want to know the probability of the height of students in this class being less than 6 ft. Again, letting X denote the height of the students in this class (recorded in units of ft), we are asking about P (X < 6), which is read as the probability of the event defined by the random variable being less than 6. We are really asking a question about the proportion of students within this class such that when their heights are measured have values less than 6 ft. The list of all students in this class is the list of all the outcomes defining the sample space, and we map a given student to the student s associated height. As a very specific example, suppose Peyton Manning is a student in the class and he is exactly 6.47 ft tall and no one else is this height. If we ask the question, what is the probability of the event that X = 6.47? Then we are really asking the question, what is the probability that Peyton Manning will randomly be selected from the class? If we ask the question, what is the probability of the event that X > 6.47? Then we are really asking, what is the probability that a student taller than Peyton 1
2 2 TROY BUTLER Manning will be randomly selected from the class? Thus, questions about the probability of rv X having certain realnumbered values are really questions about the probability of certain outcomes in the sample space. The last sentence in the above paragraph implies that if we want to determine the probability distribution of random variable X, then we must consider the underlying probability of the sample space it acts upon! How do we determine the probabilities of these various outcomes in this sample space? In what follows, we use S (read script S ) to denote the sample space and s S to denote a particular outcome (or sample) s in this sample space. Uppercase letters denote random variables and their lowercase counterparts represent particular real numbers, for example X(s) = x indicates that outcome s is mapped to real number x by rv X. 2. Discrete random variables and their distributions 2.1. Bernoulli random variables. Consider an experiment with the following two outcomes: success (S) and failure (F ). Thus, S = {S, F }. Define the rv X : S R as, X(S) = 1, and X(F ) = 0. We define a Bernoulli random variable as any rv whose only possible values are 0 and 1. A Bernoulli trial is an experiment that will result in one of two outcomes, a success or a failure. The canonical example for a Bernoulli trial is a coin toss where the coin landing heads up is a success with success probability denoted by 0 ρ 1 and landing tails up is a failure with failure probability given by 1 ρ. The pmf for Bernoulli rv X : {S, F } {0, 1} is given as above with p(1) = ρ and p(0) = 1 ρ. We often denote X Bernoulli(ρ) to indicate that rv X has a Bernoulli distribution with success probability ρ. Bernoulli rv s and the concept of independent identically distributed (or i.i.d. or iid) Bernoulli trials is critical in many areas of probability theory including the development of the Binomial distribution. Any rv (continuous or discrete) X can be used to define a Bernoulli rv simply by identifying an event of interest. For example, we can let X denote the price paid by all firsttime home buyers in the greater Denver area. Clearly X is not a Bernoulli rv as there are lots of prices that could be paid. However, if we decide that we are interested only in determining the probability that firsttime home buyers paid less than $250,000, now we have defined a brandspankingnew Bernoulli rv that we call Y (since X is already taken). Here, Y is really a function of X and since X is a function on the sample space defined by firsttime home buyers, so is Y. If X <$250,000, then Y = 1, otherwise Y = 0. The probability of success is defined by P (X < $250, 000). All that is necessary to define a Bernoulli rv is to somehow define a rule that separates the sample space into two disjoint sets where one of those sets gets mapped to 1 and the other gets mapped to 0.
3 STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE Binomial random variables. Let X be the sum of n i.i.d. (independent identically distributed) Bernoulli trials with success probability ρ, then X Binomial(n, ρ) with pmf: b(x; n, ρ) := n x ρ x (1 ρ) n x x {0, 1, 2,..., n} 0 otherwise What does S look like? Suppose there are 3 Bernoulli trials defining the sample space, then S := {SSS, SSF, SF S, F SS, SF F, F SF, F F S, F F F } defines all of the possible 8 distinct outcomes from the experiment. The rv X maps s S to the number of S s showing up in the element s (keep the s s straight here). For example, if s = SSS then X(s) = 3, if s = SF S then X(s) = 2 but s = SSF also has X(s) = 2 because the rv X does not care which order the S s appear but only the number of them (that is the rule that defines X). We use B(x; n, p) to denote the cdf of a binomial rv X. This does not give the probability of X = x (that is given by P (x) which is a shorthand way of denoting the pmf evaluated at x), it gives the probability of the event X x. Given a dichotomous population (meaning a population defined by two disjoint sets satisfying some rule ) of size N, if we use a sample of n from this population without replacement, then the rv X counting the number of successes in the n samples is not a binomial distribution. Why? Each trial within the experiment is not independent. However, if n/n < 0.05, then we can reasonably approximate the distribution of X as a binomial distribution. In the example of firsttime home buyers, if we say that we randomly sample 8 names from a list of firsttime home buyers (and assume this list has N names so that 8/N < 0.05), and we want to know the probability that at least 3 of them paid less than $250,000, then we are asking a question about a rv that has a binomial distribution with n = 8 and probability of success given by P (X < $250, 000) where X is the price paid as described previously. This new rv can be called Y (but if you decide to list the intermediate step of defining a Bernoulli rv and use Y to denote this associated Bernoulli rv as was done previously, then you should call the binomial rv something else like W to avoid confusion) Poisson random variables. The Poisson distribution is used to describe the probabilities of x numbers of events occurring in a fixed interval of time or space where λ represents the mean frequency per unit time/space. For example, the number of cars passing through an intersection in a fixed unit of time, the number of phone calls being routed through a cell tower in a given hour, or the number of chocolate chips per cookie baked from a big batch are often appropriately modeled by Poisson random variables.
4 4 TROY BUTLER A random variable X follows the Poisson distribution with parameter λ (λ > 0) if the pmf of X is given by e λ λ x x! x {0, 1, 2, 3,...} p(x; λ) = 0 otherwise. Remark 1. Given a binomial pmf b(x; n, p), if we let n and p 0 s.t. np λ > 0, then b(x; n, p) p(x; λ). The above remark implies that even though the binomial distribution might be the correct distribution to model the specific problem you are considering, it might be more computationally practical to use a Poisson distribution to approximate the answers. However, this approximation only holds in certain cases and we use the rule of thumb that this approximation holds when n > 50 and nρ < 5. In this case, we approximate the binomial distribution with the Poisson distribution where λ = nρ. Theorem 1. If the number of events that can occur in a time interval are independent with a mean rate λ and there are t disjoint time intervals, then X = the number of events occurring in the t time intervals follows a Poisson distribution with mean λt. Returning again to the example of first time homebuyers, we might want to model the number of firsttime home buyers in any year. We would have to know or be given data over the years in which to estimate the mean number of firsttime home buyers to use as the parameter in the Poisson distribution. Suppose we have such a model distribution and the mean number of firsttime home buyers in any 12 month span is 24,000, and we now want to model the number of firsttime home buyers in any 6 month span, then it is reasonable to take a Poisson distribution with parameter 12,000 (by the above theorem) Nonnamed distributions. When given a description of a finite (or countable) sample space and a rv X that does not conform to the type of descriptions that the named distributions above model, we must use the description along with rules of probabilitiy/logic/etc. to determine the distribution of X (meaning we must determine what the pmf is). 3. Continuous random variables and their distributions The common continuous distributions used in this class are the uniform, exponential, and normal/student T distributions. It will almost always be immediately clear from context which one applies as terms like uniform or equally likely show up when describing the uniform distribution and you will almost always be told whether or not the exponential or normal distribution is used to model the distribution of a particular rv. The exception is when we consider statistics (quick: what is a statistic?). Specifically, we often look at sample means or sample proportions as statistics and with a large enough sample size, the distributions of these statistics are approximately normal (Student T is approximately normal) by the Central Limit
5 STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE 5 Theorem (CLT). You will know which distribution to use in these cases based on the sample size and the use of either the exact or approximate standard deviation as we discuss in chapter 7.
3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
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
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 35, 36 Special discrete random variable distributions we will cover
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
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 informationRandom Variable: A function that assigns numerical values to all the outcomes in the sample space.
STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.
More 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 information5. Continuous Random Variables
5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be
More 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 informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More 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 informationBinomial random variables
Binomial and Poisson Random Variables Solutions STATUB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance
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 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 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 informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More 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 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 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 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 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 informationST 371 (IV): Discrete Random Variables
ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible
More informationP(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
More informationStats on the TI 83 and TI 84 Calculator
Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and
More information1.1 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 20092016 by Jenny A. Baglivo. All Rights Reserved. Contents 1 MATH4427 Notebook 1 3 1.1 Introduction, and Review of Probability
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 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 informationPractice 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:
More informatione.g. arrival of a customer to a service station or breakdown of a component in some system.
Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be
More informationExponential Distribution
Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1
More informationThe Exponential Distribution
21 The Exponential Distribution From DiscreteTime to ContinuousTime: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding
More informationCHAPTER 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,
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in reallife applications that they have been given their own names.
More informationAggregate Loss Models
Aggregate Loss Models Chapter 9 Stat 477  Loss Models Chapter 9 (Stat 477) Aggregate Loss Models Brian Hartman  BYU 1 / 22 Objectives Objectives Individual risk model Collective risk model Computing
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 informationECE302 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
More informationPrinciple of Data Reduction
Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 41/42 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationChapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams
Review for Final Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Histogram Boxplots Chapter 3: Set
More informationStatistics 100A Homework 4 Solutions
Problem 1 For a discrete random variable X, Statistics 100A Homework 4 Solutions Ryan Rosario Note that all of the problems below as you to prove the statement. We are proving the properties of epectation
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationMATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...
MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 20092016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................
More informationThe normal approximation to the binomial
The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More information6.2. 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
More informationLECTURE 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
More informationChapter 9 Monté Carlo Simulation
MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?
More informationExample 1: Dear Abby. Stat Camp for the Fulltime MBA Program
Stat Camp for the Fulltime MBA Program Daniel Solow Lecture 4 The Normal Distribution and the Central Limit Theorem 188 Example 1: Dear Abby You wrote that a woman is pregnant for 266 days. Who said so?
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 37, 38 The remaining discrete random
More informationPractice 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 multiplechoice
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
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 informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 101634501 Probability and Statistics for Engineers Winter 20102011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete
More informationChapter 5. Discrete Probability Distributions
Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable
More informationON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME
ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME Alexey Chuprunov Kazan State University, Russia István Fazekas University of Debrecen, Hungary 2012 Kolchin s generalized allocation scheme A law of
More informationProbability distributions
Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.142.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,
More informationThe normal approximation to the binomial
The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There
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 informationProbability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.
Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111
More informationBinomial random variables (Review)
Poisson / Empirical Rule Approximations / Hypergeometric Solutions STATUB.3 Statistics for Business Control and Regression Models Binomial random variables (Review. Suppose that you are rolling a die
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 informationSection 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
More information2 Binomial, Poisson, Normal Distribution
2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability
More informationLecture 5 : The Poisson Distribution
Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,
More informationHomework 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 21 Since there can be anywhere from 0 to 4 aces, the
More information4: 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
More informationProbability Generating Functions
page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
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 informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationTEST 2 STUDY GUIDE. 1. Consider the data shown below.
2006 by The Arizona Board of Regents for The University of Arizona All rights reserved Business Mathematics I TEST 2 STUDY GUIDE 1 Consider the data shown below (a) Fill in the Frequency and Relative Frequency
More informationSTAT 35A HW2 Solutions
STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },
More informationMAT 211 Introduction to Business Statistics I Lecture Notes
MAT 211 Introduction to Business Statistics I Lecture Notes Muhammad ElTaha Department of Mathematics and Statistics University of Southern Maine 96 Falmouth Street Portland, ME 041049300 MAT 211, Spring
More informationLecture 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
More informationCombinatorics: 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
More informationFinal 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
More informationHypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...
Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................
More informationThe 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
More informationSTAT 830 Convergence in Distribution
STAT 830 Convergence in Distribution Richard Lockhart Simon Fraser University STAT 830 Fall 2011 Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 Fall 2011 1 / 31
More informationAn 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
More information6 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
More informationProbability 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,chisquared distributions, normal approx'n to the binomial Uniform [,1] random
More informationChapter 3 Random Variables and Probability Distributions
Math 322 Probabilit and Statistical Methods Chapter 3 Random Variables and Probabilit Distributions In statistics we deal with random variables variables whose observed value is determined b chance. Random
More informationMath 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,
More information4. 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
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 information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationYou 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.
More informationSampling Central Limit Theorem Proportions. Outline. 1 Sampling. 2 Central Limit Theorem. 3 Proportions
Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Populations and samples When we use statistics, we are trying to find out information about
More informationProbability Calculator
Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that
More informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationChapter 5: Normal Probability Distributions  Solutions
Chapter 5: Normal Probability Distributions  Solutions Note: All areas and zscores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that
More informationLecture 10: Depicting Sampling Distributions of a Sample Proportion
Lecture 10: Depicting Sampling Distributions of a Sample Proportion Chapter 5: Probability and Sampling Distributions 2/10/12 Lecture 10 1 Sample Proportion 1 is assigned to population members having a
More informationIntroduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 5.1 Homework Answers
Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 5.1 Homework Answers 5.18 Attitudes toward drinking and behavior studies. Some of the methods in this section are approximations
More information0 x = 0.30 x = 1.10 x = 3.05 x = 4.15 x = 6 0.4 x = 12. f(x) =
. A mailorder 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
More information2WB05 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
More informationAPPLICATION OF LINEAR REGRESSION MODEL FOR POISSON DISTRIBUTION IN FORECASTING
APPLICATION OF LINEAR REGRESSION MODEL FOR POISSON DISTRIBUTION IN FORECASTING Sulaimon Mutiu O. Department of Statistics & Mathematics Moshood Abiola Polytechnic, Abeokuta, Ogun State, Nigeria. Abstract
More informationExploratory Data Analysis
Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
More information