Chapter 1: Stochastic Processes
|
|
- Kelley Hampton
- 6 years ago
- Views:
Transcription
1 Chater 1: Stochastic Processes 4 What are Stochastic Processes, and how do they fit in? STATS 210 Foundations of Statistics and Probability Tools for understanding randomness (random variables, distributions) STATS 310 Statistics Randomness in Pattern STATS 325 Probability Randomness in Process Stats 210: laid the foundations of both Statistics and Probability: the tools for understanding randomness. Stats 310: develos the theory for understanding randomness in attern: tools for estimating arameters(maximum likelihood), testing hyotheses, modelling atterns in data (regression models). Stats 325: develos the theory for understanding randomness in rocess. A rocess is a sequence of events where each ste follows from the last after a random choice. What sort of roblems will we cover in Stats 325? Here are some examles of the sorts of roblems that we study in this course. Gambler s Ruin You start with $30 and toss a fair coin reeatedly. Every time you throw a Head, you win $5. Every time you throw a Tail, you lose $5. You will sto when you reach $100 or when you lose everything. What is the robability that you lose everything? Answer: 70%.
2
3 5 Winning at tennis What is your robability of winning a game of tennis, starting from the even score Deuce (40-40), if your robability of winning each oint is 0.3 and your oonent s is 0.7? Answer: 15%. DEUCE (D) q q VENUS AHEAD (A) VENUS BEHIND (B) q VENUS WINS (W) VENUS LOSES (L) Winning a lottery A million eole have bought tickets for the weekly lottery draw. Each erson has a robability of one-in-a-million of selecting the winning numbers. If more than one erson selects the winning numbers, the winner will be chosen at random from all those with matching numbers. You watch the lottery draw on TV and your numbers match the winning numbers!!! Only a one-in-a-million chance, and there were only a million layers, so surely you will win the rize? Not quite... What is the robability you will win? Answer: only 63%. Drunkard s walk A very drunk erson staggers to left and right as he walks along. With each ste he takes, he staggers one ace to the left with robability 0.5, and one ace to the right with robability 0.5. What is the exected number of aces he must take before he ends u one ace to the left of his starting oint? Arrived! Answer: the exectation is infinite!
4 6 Pyramid selling schemes Have you received a chain letter like this one? Just send $10 to the erson whose name comes at the to of the list, and add your own name to the bottom of the list. Send the letter to as many eole as you can. Within a few months, the letter romises, you will have received $77,000 in $10 notes! Will you? Answer: it deends uon the resonse rate. However, with a fairly realistic assumtion about resonse rate, we can calculate an exected return of $76 with a 64% chance of getting nothing! Note: Pyramid selling schemes like this are rohibited under the Fair Trading Act, and it is illegal to articiate in them. Sread of SARS The figure to the right shows the sread of the disease SARS (Severe Acute Resiratory Syndrome) through Singaore in With this attern of infections, what is the robability that the disease eventually dies out of its own accord? Answer:
5 7 Markov s Marvellous Mystery Tours Mr Markov s Marvellous Mystery Tours romises an All-Stochastic Tourist Exerience for the town of Rotorua. Mr Markov has eight tourist attractions, to which he will take his clients comletely at random with the robabilities shown below. He romises at least three exciting attractions er tour, ending at either the Lady Knox Geyser or the Tarawera Volcano. (Unfortunately he makes no mention of how the haless tourist might get home from these laces.) What is the exected number of activitiesfor a tour startingfrom the museum? 2. Cruise 4. Flying Fox 1 6. Geyser 1 1. Museum 3. Buried Village 5. Hangi 7. Helicoter 8. Volcano 1 1 Answer: 4.2. Structure of the course Probability. Probability and random variables, with secial focus on conditional robability. Finding hitting robabilities for stochastic rocesses. Exectation. Exectation and variance. Introduction to conditional exectation, and its alication in finding exected reaching times in stochastic rocesses. Generating functions. Introduction to robability generating functions, and their alications to stochastic rocesses, esecially the Random Walk. Branching rocess. This rocess is a simle model for reroduction. Examles are the yramid selling scheme and the sread of SARS above.
6 8 Markov chains. Almost all the examles we look at throughout the course can be formulated as Markov chains. By develoing a single unifying theory, we can easily tackle comlex roblems with many states and transitions like Markov s Marvellous Mystery Tours above. The rest of this chater covers: quick revision of samle saces and random variables; formal definition of stochastic rocesses. 1.1 Revision: Samle saces and random variables Definition: A random exeriment is a hysical situation whose outcome cannot be redicted until it is observed. Definition: A samle sace, Ω, is a set of ossible outcomes of a random exeriment. Examle: Random exeriment: Toss a coin once. Samle sace: Ω ={head, tail} Definition: A random variable, X, is defined as a function from the samle sace to the real numbers: X : Ω R. That is, a random variable assigns a real number to every ossible outcome of a random exeriment. Examle: Random exeriment: Toss a coin once. Samle sace: Ω = {head, tail}. An examle of a random variable: X : Ω R mas head 1, tail 0. Essential oint: A random variable is a way of roducing random real numbers.
7 9 1.2 Stochastic Processes Definition: A stochastic rocess is a family of random variables, {X(t) : t T}, where t usually denotes time. That is, at every time t in the set T, a random number X(t) is observed. Definition: {X(t) : t T} is a discrete-time rocess if the set T is finite or countable. In ractice, this generally means T = {0,1,2,3,...} Thus a discrete-timerocess is {X(0),X(1),X(2),X(3),...}: a new random number recorded at every time 0, 1, 2, 3,... Definition: {X(t) : t T} is a continuous-time rocess if T is not finite or countable. In ractice, this generally means T = [0, ), or T = [0,K] for some K. Thus a continuous-timerocess {X(t) : t T} has a random number X(t) recorded at every instant in time. (Note that X(t) need not change at every instant in time, but it is allowed to change at any time; i.e. not just at t = 0,1,2,..., like a discrete-time rocess.) Definition: The state sace, S, is the set of real values that X(t) can take. Every X(t) takes a value in R, but S will often be a smaller set: S R. For examle, if X(t) is the outcome of a coin tossed at time t, then the state sace is S = {0,1}. Definition: The state sace S is discrete if it is finite or countable. Otherwise it is continuous. The state sace S is the set of states that the stochastic rocess can be in.
8 10 For Reference: Discrete Random Variables 1. Binomial distribution Notation: X Binomial(n, ). Descrition: number of successes in n indeendent trials, each with robability of success. Probability function: Mean: E(X) = n. f X (x) = P(X = x) = ( ) n x (1 ) n x x Variance: Var(X) = n(1 ) = nq, where q = 1. for x = 0,1,...,n. Sum: If X Binomial(n,), Y Binomial(m,), and X and Y are indeendent, then X +Y Bin(n+m, ). 2. Poisson distribution Notation: X Poisson(λ). Descrition: arises out of the Poisson rocess as the number of events in a fixed time or sace, when events occur at a constant average rate. Also used in many other situations. Probability function: f X (x) = P(X = x) = λx x! e λ for x = 0,1,2,... Mean: E(X) = λ. Variance: Var(X) = λ. Sum: If X Poisson(λ), Y Poisson(µ), and X and Y are indeendent, then X +Y Poisson(λ+µ).
9 11 3. Geometric distribution Notation: X Geometric(). Descrition: number of failures before the first success in a sequence of indeendent trials, each with P(success) =. Probability function: f X (x) = P(X = x) = (1 ) x for x = 0,1,2,... Mean: E(X) = 1 = q, where q = 1. Variance: Var(X) = 1 2 = q 2, where q = 1. Sum: if X 1,...,X k are indeendent, and each X i Geometric(), then X X k Negative Binomial(k,). 4. Negative Binomial distribution Notation: X NegBin(k, ). Descrition: number of failures before the kth success in a sequence of indeendent trials, each with P(success) =. Probability function: ( ) k +x 1 f X (x) = P(X = x) = k (1 ) x for x = 0,1,2,... x Mean: E(X) = k(1 ) = kq, where q = 1. Variance: Var(X) = k(1 ) 2 = kq 2, where q = 1. Sum: IfX NegBin(k, ),Y NegBin(m, ),andx andy areindeendent, then X +Y NegBin(k +m, ).
10 12 5. Hyergeometric distribution Notation: X Hyergeometric(N, M, n). Descrition: Samling without relacement from a finite oulation. Given N objects, of which M are secial. Draw n objects without relacement. X is the number of the n objects that are secial. Probability function: f X (x) = P(X = x) = Mean: E(X) = n, where = M N. ( M N M ) x)( n x ( N for n) ( N n ) Variance: Var(X) = n(1 ), where = M N 1 N. { x = max(0, n+m N) to x = min(n, M). 6. Multinomial distribution Notation: X = (X 1,...,X k ) Multinomial(n; 1, 2,..., k ). Descrition: there are n indeendent trials, each with k ossible outcomes. Let i = P(outcome i) for i = 1,...k. Then X = (X 1,...,X k ), where X i is the number of trials with outcome i, for i = 1,...,k. Probability function: n! f X (x) = P(X 1 = x 1,...,X k = x k ) = x 1!...x k! x 1 1 x x k k k k for x i {0,...,n} i with x i = n, and where i 0 i, i = 1. Marginal distributions: X i Binomial(n, i ) for i = 1,...,k. i=1 Mean: E(X i ) = n i for i = 1,...,k. Variance: Var(X i ) = n i (1 i ), for i = 1,...,k. Covariance: cov(x i,x j ) = n i j, for all i j. i=1
11 13 Continuous Random Variables 1. Uniform distribution Notation: X Uniform(a, b). Probability density function (df): f X (x) = 1 b a for a < x < b. Cumulative distribution function: Mean: E(X) = a+b 2. Variance: Var(X) = (b a)2. 12 F X (x) = P(X x) = x a for a < x < b. b a F X (x) = 0 for x a, and F X (x) = 1 for x b. 2. Exonential distribution Notation: X Exonential(λ). Probability density function (df): f X (x) = λe λx for 0 < x <. Cumulative distribution function: F X (x) = P(X x) = 1 e λx for 0 < x <. F X (x) = 0 for x 0. Mean: E(X) = 1 λ. Variance: Var(X) = 1 λ 2. Sum: if X 1,...,X k are indeendent, and each X i Exonential(λ), then X X k Gamma(k,λ).
12 14 3. Gamma distribution Notation: X Gamma(k, λ). Probability density function (df): f X (x) = λk Γ(k) xk 1 e λx for 0 < x <, where Γ(k) = 0 y k 1 e y dy (the Gamma function). Cumulative distribution function: no closed form. Mean: E(X) = k λ. Variance: Var(X) = k λ 2. Sum: if X 1,...,X n are indeendent, and X i Gamma(k i, λ), then X X n Gamma(k k n, λ). 4. Normal distribution Notation: X Normal(µ, σ 2 ). Probability density function (df): f X (x) = 1 2πσ 2 e{ (x µ)2 /2σ 2 } for < x <. Cumulative distribution function: no closed form. Mean: E(X) = µ. Variance: Var(X) = σ 2. Sum: if X 1,...,X n are indeendent, and X i Normal(µ i, σi 2 ), then X X n Normal(µ µ n, σ σ 2 n).
13 15 Uniform(a, b) Probability Density Functions 1 b a f X (x) a b x Exonential(λ) λ = 2 λ = 1 Gamma(k, λ) k = 2, λ = 1 k = 2, λ = 0.3 Normal(µ, σ 2 ) σ = 2 σ = 4 µ
Lecture 2. Main Topics: (Part II) Chapter 2 (2-7), Chapter 3. Bayes Theorem: Let A, B be two events, then. The probabilities P ( B), probability of B.
STT315, Section 701, Summer 006 Lecture (Part II) Main Toics: Chater (-7), Chater 3. Bayes Theorem: Let A, B be two events, then B A) = A B) B) A B) B) + A B) B) The robabilities P ( B), B) are called
More informationA random variable X is a function that assigns (real) numbers to the elements of the sample space S of a random experiment.
RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS A random variable X is a function that assigns (real) numbers to the elements of the samle sace S of a random exeriment. The value sace V of a random variable
More informationMidterm Exam: Tuesday 28 March in class Sample exam problems ( Homework 5 ) available tomorrow at the latest
Plan Martingales 1. Basic Definitions 2. Examles 3. Overview of Results Reading: G&S Section 12.1-12.4 Next Time: More Martingales Midterm Exam: Tuesday 28 March in class Samle exam roblems ( Homework
More informationTOPIC: PROBABILITY DISTRIBUTIONS
TOPIC: PROBABILITY DISTRIBUTIONS There are two types of random variables: A Discrete random variable can take on only specified, distinct values. A Continuous random variable can take on any value within
More information***SECTION 7.1*** Discrete and Continuous Random Variables
***SECTION 7.*** Discrete and Continuous Random Variables Samle saces need not consist of numbers; tossing coins yields H s and T s. However, in statistics we are most often interested in numerical outcomes
More informationPolicyholder Outcome Death Disability Neither Payout, x 10,000 5, ,000
Two tyes of Random Variables: ) Discrete random variable has a finite number of distinct outcomes Examle: Number of books this term. ) Continuous random variable can take on any numerical value within
More informationand their probabilities p
AP Statistics Ch. 6 Notes Random Variables A variable is any characteristic of an individual (remember that individuals are the objects described by a data set and may be eole, animals, or things). Variables
More information(Practice Version) Midterm Exam 1
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran September 19, 2014 (Practice Version) Midterm Exam 1 Last name First name SID Rules. DO NOT open
More informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 3: Special Discrete Random Variable Distributions Section 3.5 Discrete Uniform Section 3.6 Bernoulli and Binomial Others sections
More informationSTA 6166 Fall 2007 Web-based Course. Notes 10: Probability Models
STA 6166 Fall 2007 Web-based Course 1 Notes 10: Probability Models We first saw the normal model as a useful model for the distribution of some quantitative variables. We ve also seen that if we make a
More informationOrdering a deck of cards... Lecture 3: Binomial Distribution. Example. Permutations & Combinations
Ordering a dec of cards... Lecture 3: Binomial Distribution Sta 111 Colin Rundel May 16, 2014 If you have ever shuffled a dec of cards you have done something no one else has ever done before or will ever
More informationUQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions.
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions. Random Variables 2 A random variable X is a numerical (integer, real, complex, vector etc.) summary of the outcome of the random experiment.
More informationProbability Theory. Probability and Statistics for Data Science CSE594 - Spring 2016
Probability Theory Probability and Statistics for Data Science CSE594 - Spring 2016 What is Probability? 2 What is Probability? Examples outcome of flipping a coin (seminal example) amount of snowfall
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More informationPart 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?
1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard
More informationSimple Random Sample
Simple Random Sample A simple random sample (SRS) of size n consists of n elements from the population chosen in such a way that every set of n elements has an equal chance to be the sample actually selected.
More informationObjectives. 3.3 Toward statistical inference
Objectives 3.3 Toward statistical inference Poulation versus samle (CIS, Chater 6) Toward statistical inference Samling variability Further reading: htt://onlinestatbook.com/2/estimation/characteristics.html
More informationRandom Variables Handout. Xavier Vilà
Random Variables Handout Xavier Vilà Course 2004-2005 1 Discrete Random Variables. 1.1 Introduction 1.1.1 Definition of Random Variable A random variable X is a function that maps each possible outcome
More informationDr. Maddah ENMG 625 Financial Eng g II 10/16/06. Chapter 11 Models of Asset Dynamics (1)
Dr Maddah ENMG 65 Financial Eng g II 0/6/06 Chater Models of Asset Dynamics () Overview Stock rice evolution over time is commonly modeled with one of two rocesses: The binomial lattice and geometric Brownian
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationChapter 8. Variables. Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 Random Variables Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc. 8.1 What is a Random Variable? Random Variable: assigns a number to each outcome of a random circumstance, or,
More information2/20/2013. of Manchester. The University COMP Building a yes / no classifier
COMP4 Lecture 6 Building a yes / no classifier Buildinga feature-basedclassifier Whatis a classifier? What is an information feature? Building a classifier from one feature Probability densities and the
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationSection Random Variables and Histograms
Section 3.1 - Random Variables and Histograms Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More information6 If and then. (a) 0.6 (b) 0.9 (c) 2 (d) Which of these numbers can be a value of probability distribution of a discrete random variable
1. A number between 0 and 1 that is use to measure uncertainty is called: (a) Random variable (b) Trial (c) Simple event (d) Probability 2. Probability can be expressed as: (a) Rational (b) Fraction (c)
More informationProbability Models.S2 Discrete Random Variables
Probability Models.S2 Discrete Random Variables Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard Results of an experiment involving uncertainty are described by one or more random
More informationReview for Final Exam Spring 2014 Jeremy Orloff and Jonathan Bloom
Review for Final Exam 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom THANK YOU!!!! JON!! PETER!! RUTHI!! ERIKA!! ALL OF YOU!!!! Probability Counting Sets Inclusion-exclusion principle Rule of product
More informationExamples: Random Variables. Discrete and Continuous Random Variables. Probability Distributions
Random Variables Examples: Random variable a variable (typically represented by x) that takes a numerical value by chance. Number of boys in a randomly selected family with three children. Possible values:
More information1 < = α σ +σ < 0. Using the parameters and h = 1/365 this is N ( ) = If we use h = 1/252, the value would be N ( ) =
Chater 6 Value at Risk Question 6.1 Since the rice of stock A in h years (S h ) is lognormal, 1 < = α σ +σ < 0 ( ) P Sh S0 P h hz σ α σ α = P Z < h = N h. σ σ (1) () Using the arameters and h = 1/365 this
More informationDiscrete Random Variables and Probability Distributions
Chapter 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables A quantity resulting from an experiment that, by chance, can assume different values. A random variable is a variable
More informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
More informationProbability mass function; cumulative distribution function
PHP 2510 Random variables; some discrete distributions Random variables - what are they? Probability mass function; cumulative distribution function Some discrete random variable models: Bernoulli Binomial
More informationDrunken Birds, Brownian Motion, and Other Random Fun
Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability
More informationIEOR 165 Lecture 1 Probability Review
IEOR 165 Lecture 1 Probability Review 1 Definitions in Probability and Their Consequences 1.1 Defining Probability A probability space (Ω, F, P) consists of three elements: A sample space Ω is the set
More informationChapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
More informationStatistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
More informationHave you ever wondered whether it would be worth it to buy a lottery ticket every week, or pondered on questions such as If I were offered a choice
Section 8.5: Expected Value and Variance Have you ever wondered whether it would be worth it to buy a lottery ticket every week, or pondered on questions such as If I were offered a choice between a million
More informationcontinuous rv Note for a legitimate pdf, we have f (x) 0 and f (x)dx = 1. For a continuous rv, P(X = c) = c f (x)dx = 0, hence
continuous rv Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a b, P(a X b) = b a f (x)dx.
More informationChapter 5 Discrete Probability Distributions. Random Variables Discrete Probability Distributions Expected Value and Variance
Chapter 5 Discrete Probability Distributions Random Variables Discrete Probability Distributions Expected Value and Variance.40.30.20.10 0 1 2 3 4 Random Variables A random variable is a numerical description
More informationStatistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 5 Discrete Probability Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 014 Pearson Education, Inc. Chap 5-1 Learning
More informationINDEX NUMBERS. Introduction
INDEX NUMBERS Introduction Index numbers are the indicators which reflect changes over a secified eriod of time in rices of different commodities industrial roduction (iii) sales (iv) imorts and exorts
More informationDiscrete Probability Distributions
Page 1 of 6 Discrete Probability Distributions In order to study inferential statistics, we need to combine the concepts from descriptive statistics and probability. This combination makes up the basics
More informationEcon 6900: Statistical Problems. Instructor: Yogesh Uppal
Econ 6900: Statistical Problems Instructor: Yogesh Uppal Email: yuppal@ysu.edu Lecture Slides 4 Random Variables Probability Distributions Discrete Distributions Discrete Uniform Probability Distribution
More informationDiscrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)
3 Discrete Random Variables and Probability Distributions Stat 4570/5570 Based on Devore s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer
More informationAn Introduction to Stochastic Calculus
An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 5 Haijun Li An Introduction to Stochastic Calculus Week 5 1 / 20 Outline 1 Martingales
More informationLecture Notes 6. Assume F belongs to a family of distributions, (e.g. F is Normal), indexed by some parameter θ.
Sufficient Statistics Lecture Notes 6 Sufficiency Data reduction in terms of a particular statistic can be thought of as a partition of the sample space X. Definition T is sufficient for θ if the conditional
More informationAsian Economic and Financial Review A MODEL FOR ESTIMATING THE DISTRIBUTION OF FUTURE POPULATION. Ben David Nissim.
Asian Economic and Financial Review journal homeage: htt://www.aessweb.com/journals/5 A MODEL FOR ESTIMATING THE DISTRIBUTION OF FUTURE POPULATION Ben David Nissim Deartment of Economics and Management,
More informationCentral Limit Theorem 11/08/2005
Central Limit Theorem 11/08/2005 A More General Central Limit Theorem Theorem. Let X 1, X 2,..., X n,... be a sequence of independent discrete random variables, and let S n = X 1 + X 2 + + X n. For each
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationObjectives. 5.2, 8.1 Inference for a single proportion. Categorical data from a simple random sample. Binomial distribution
Objectives 5.2, 8.1 Inference for a single roortion Categorical data from a simle random samle Binomial distribution Samling distribution of the samle roortion Significance test for a single roortion Large-samle
More informationBinomial formulas: The binomial coefficient is the number of ways of arranging k successes among n observations.
Chapter 8 Notes Binomial and Geometric Distribution Often times we are interested in an event that has only two outcomes. For example, we may wish to know the outcome of a free throw shot (good or missed),
More information2017 Fall QMS102 Tip Sheet 2
Chapter 5: Basic Probability 2017 Fall QMS102 Tip Sheet 2 (Covering Chapters 5 to 8) EVENTS -- Each possible outcome of a variable is an event, including 3 types. 1. Simple event = Described by a single
More informationBinomial and multinomial distribution
1-Binomial distribution Binomial and multinomial distribution The binomial probability refers to the probability that a binomial experiment results in exactly "x" successes. The probability of an event
More informationCS134: Networks Spring Random Variables and Independence. 1.2 Probability Distribution Function (PDF) Number of heads Probability 2 0.
CS134: Networks Spring 2017 Prof. Yaron Singer Section 0 1 Probability 1.1 Random Variables and Independence A real-valued random variable is a variable that can take each of a set of possible values in
More informationCentral Limit Thm, Normal Approximations
Central Limit Thm, Normal Approximations Engineering Statistics Section 5.4 Josh Engwer TTU 23 March 2016 Josh Engwer (TTU) Central Limit Thm, Normal Approximations 23 March 2016 1 / 26 PART I PART I:
More information1/2 2. Mean & variance. Mean & standard deviation
Question # 1 of 10 ( Start time: 09:46:03 PM ) Total Marks: 1 The probability distribution of X is given below. x: 0 1 2 3 4 p(x): 0.73? 0.06 0.04 0.01 What is the value of missing probability? 0.54 0.16
More informationSection M Discrete Probability Distribution
Section M Discrete Probability Distribution A random variable is a numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are typically denoted
More informationShifting our focus. We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why?
Probability Introduction Shifting our focus We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why? What is Probability? Probability is used
More informationII - Probability. Counting Techniques. three rules of counting. 1multiplication rules. 2permutations. 3combinations
II - Probability Counting Techniques three rules of counting 1multiplication rules 2permutations 3combinations Section 2 - Probability (1) II - Probability Counting Techniques 1multiplication rules In
More informationLecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
More informationChapter 5. Discrete Probability Distributions. McGraw-Hill, Bluman, 7 th ed, Chapter 5 1
Chapter 5 Discrete Probability Distributions McGraw-Hill, Bluman, 7 th ed, Chapter 5 1 Chapter 5 Overview Introduction 5-1 Probability Distributions 5-2 Mean, Variance, Standard Deviation, and Expectation
More informationChapter 7: Random Variables
Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables 7.2 Means and Variances of Random Variables 1 Introduction A random variable is a function that associates a unique numerical value
More informationLecture 23: April 10
CS271 Randomness & Computation Spring 2018 Instructor: Alistair Sinclair Lecture 23: April 10 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationMATH/STAT 3360, Probability FALL 2013 Toby Kenney
MATH/STAT 3360, Probability FALL 2013 Toby Kenney In Class Examples () September 6, 2013 1 / 92 Basic Principal of Counting A statistics textbook has 8 chapters. Each chapter has 50 questions. How many
More informationLESSON 9: BINOMIAL DISTRIBUTION
LESSON 9: Outline The context The properties Notation Formula Use of table Use of Excel Mean and variance 1 THE CONTEXT An important property of the binomial distribution: An outcome of an experiment is
More informationChapter 4: Commonly Used Distributions. Statistics for Engineers and Scientists Fourth Edition William Navidi
Chapter 4: Commonly Used Distributions Statistics for Engineers and Scientists Fourth Edition William Navidi 2014 by Education. This is proprietary material solely for authorized instructor use. Not authorized
More informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 4: Special Discrete Random Variable Distributions Sections 3.7 & 3.8 Geometric, Negative Binomial, Hypergeometric NOTE: The discrete
More informationChapter 3. Discrete Probability Distributions
Chapter 3 Discrete Probability Distributions 1 Chapter 3 Overview Introduction 3-1 The Binomial Distribution 3-2 Other Types of Distributions 2 Chapter 3 Objectives Find the exact probability for X successes
More information5.2 Random Variables, Probability Histograms and Probability Distributions
Chapter 5 5.2 Random Variables, Probability Histograms and Probability Distributions A random variable (r.v.) can be either continuous or discrete. It takes on the possible values of an experiment. It
More information4-1. Chapter 4. Commonly Used Distributions by The McGraw-Hill Companies, Inc. All rights reserved.
4-1 Chapter 4 Commonly Used Distributions 2014 by The Companies, Inc. All rights reserved. Section 4.1: The Bernoulli Distribution 4-2 We use the Bernoulli distribution when we have an experiment which
More informationChapter 4 and 5 Note Guide: Probability Distributions
Chapter 4 and 5 Note Guide: Probability Distributions Probability Distributions for a Discrete Random Variable A discrete probability distribution function has two characteristics: Each probability is
More informationProbability and distributions
2 Probability and distributions The concepts of randomness and probability are central to statistics. It is an empirical fact that most experiments and investigations are not perfectly reproducible. The
More informationElementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
More informationBusiness Statistics. Chapter 5 Discrete Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationChapter 5 Student Lecture Notes 5-1. Department of Quantitative Methods & Information Systems. Business Statistics
Chapter 5 Student Lecture Notes 5-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals
More informationReliability and Risk Analysis. Survival and Reliability Function
Reliability and Risk Analysis Survival function We consider a non-negative random variable X which indicates the waiting time for the risk event (eg failure of the monitored equipment, etc.). The probability
More informationBusiness Statistics Midterm Exam Fall 2013 Russell
Name SOLUTION Business Statistics Midterm Exam Fall 2013 Russell Do not turn over this page until you are told to do so. You will have 2 hours to complete the exam. There are a total of 100 points divided
More informationChapter 7: Random Variables and Discrete Probability Distributions
Chapter 7: Random Variables and Discrete Probability Distributions 7. Random Variables and Probability Distributions This section introduced the concept of a random variable, which assigns a numerical
More informationMean of a Discrete Random variable. Suppose that X is a discrete random variable whose distribution is : :
Dr. Kim s Note (December 17 th ) The values taken on by the random variable X are random, but the values follow the pattern given in the random variable table. What is a typical value of a random variable
More informationThe Normal Distribution
The Normal Distribution The normal distribution plays a central role in probability theory and in statistics. It is often used as a model for the distribution of continuous random variables. Like all models,
More informationHomework 10 Solution Section 4.2, 4.3.
MATH 00 Homewor Homewor 0 Solution Section.,.3. Please read your writing again before moving to the next roblem. Do not abbreviate your answer. Write everything in full sentences. Write your answer neatly.
More informationBinomial distribution
Binomial distribution Jon Michael Gran Department of Biostatistics, UiO MF9130 Introductory course in statistics Tuesday 24.05.2010 1 / 28 Overview Binomial distribution (Aalen chapter 4, Kirkwood and
More information5. In fact, any function of a random variable is also a random variable
Random Variables - Class 11 October 14, 2012 Debdeep Pati 1 Random variables 1.1 Expectation of a function of a random variable 1. Expectation of a function of a random variable 2. We know E(X) = x xp(x)
More informationMath 180A. Lecture 5 Wednesday April 7 th. Geometric distribution. The geometric distribution function is
Geometric distribution The geometric distribution function is x f ( x) p(1 p) 1 x {1,2,3,...}, 0 p 1 It is the pdf of the random variable X, which equals the smallest positive integer x such that in a
More informationMATH/STAT 3360, Probability FALL 2012 Toby Kenney
MATH/STAT 3360, Probability FALL 2012 Toby Kenney In Class Examples () August 31, 2012 1 / 81 A statistics textbook has 8 chapters. Each chapter has 50 questions. How many questions are there in total
More informationMA 1125 Lecture 14 - Expected Values. Wednesday, October 4, Objectives: Introduce expected values.
MA 5 Lecture 4 - Expected Values Wednesday, October 4, 27 Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationEcon 250 Fall Due at November 16. Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling
Econ 250 Fall 2010 Due at November 16 Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling 1. Suppose a firm wishes to raise funds and there are a large number of independent financial
More informationMAS187/AEF258. University of Newcastle upon Tyne
MAS187/AEF258 University of Newcastle upon Tyne 2005-6 Contents 1 Collecting and Presenting Data 5 1.1 Introduction...................................... 5 1.1.1 Examples...................................
More informationAnnex 4 - Poverty Predictors: Estimation and Algorithm for Computing Predicted Welfare Function
Annex 4 - Poverty Predictors: Estimation and Algorithm for Comuting Predicted Welfare Function The Core Welfare Indicator Questionnaire (CWIQ) is an off-the-shelf survey ackage develoed by the World Bank
More informationStochastic Processes and Financial Mathematics (part one) Dr Nic Freeman
Stochastic Processes and Financial Mathematics (part one) Dr Nic Freeman December 15, 2017 Contents 0 Introduction 3 0.1 Syllabus......................................... 4 0.2 Problem sheets.....................................
More informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationNormal Distribution. Notes. Normal Distribution. Standard Normal. Sums of Normal Random Variables. Normal. approximation of Binomial.
Lecture 21,22, 23 Text: A Course in Probability by Weiss 8.5 STAT 225 Introduction to Probability Models March 31, 2014 Standard Sums of Whitney Huang Purdue University 21,22, 23.1 Agenda 1 2 Standard
More informationNo, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
More informationBasics of Probability
Basics of Probability By A.V. Vedpuriswar October 2, 2016 2, 2016 Random variables and events A random variable is an uncertain quantity. A outcome is an observed value of a random variable. An event is
More informationAssignment 3 - Statistics. n n! (n r)!r! n = 1,2,3,...
Assignment 3 - Statistics Name: Permutation: Combination: n n! P r = (n r)! n n! C r = (n r)!r! n = 1,2,3,... n = 1,2,3,... The Fundamental Counting Principle: If two indepndent events A and B can happen
More informationSTAT/MATH 395 PROBABILITY II
STAT/MATH 395 PROBABILITY II Distribution of Random Samples & Limit Theorems Néhémy Lim University of Washington Winter 2017 Outline Distribution of i.i.d. Samples Convergence of random variables The Laws
More informationChapter 4 Probability Distributions
Slide 1 Chapter 4 Probability Distributions Slide 2 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 4-5
More informationChapter 6: Random Variables. Ch. 6-3: Binomial and Geometric Random Variables
Chapter : Random Variables Ch. -3: Binomial and Geometric Random Variables X 0 2 3 4 5 7 8 9 0 0 P(X) 3???????? 4 4 When the same chance process is repeated several times, we are often interested in whether
More information