Discrete Distributions
|
|
- Cecil Townsend
- 5 years ago
- Views:
Transcription
1 Discrete Distributions
2 Applications of the Binomial Distribution A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing research firm receives survey responses of yes I will buy or no I will not New job applicants either accept the offer or reject it Your team either wins or loses the football game at the company picnic
3 Bernoulli Random Variable If an experiment consists of a single trial and the outcome of the trial can only be either a success * or a failure, then the trial is called a Bernoulli trial. The number of success X in one Bernoulli trial, which can be 1 or 0, is a Bernoulli random variable. Note: If p is the probability of success in a Bernoulli experiment, the E(X) = p and V(X) = p(1 p). * The terms success and failure are simply statistical terms, and do not have positive or negative implications. In a production setting, finding a defective product may be termed a success, although it is not a positive result.
4 The Binomial Random Variable Consider a Bernoulli Process in which we have a sequence of n identical trials satisfying the following conditions: 1. Each trial has two possible outcomes, called success *and failure. The two outcomes are mutually exclusive and exhaustive.. The probability of success, denoted by p, remains constant from trial to trial. The probability of failure is denoted by q, where q = 1-p. 3. The n trials are independent. That is, the outcome of any trial does not affect the outcomes of the other trials. A random variable, X, that counts the number of successes in n Bernoulli trials, where p is the probability of success* in any given trial, is said to follow the binomial probability distribution with parameters n (number of trials) and p (probability of success). We call X the binomial random variable. * The terms success and failure are simply statistical terms, and do not have positive or negative implications. In a production setting, finding a defective product may be termed a success, although it is not a positive result.
5 The Binomial Distribution: Properties A fixed number of observations, n ex. 15 tosses of a coin; ten light bulbs taken from a warehouse Two mutually exclusive and collectively exhaustive categories ex. head or tail in each toss of a coin; defective or not defective light bulb; having a boy or girl Generally called success and failure Probability of success is p, probability of failure is 1 p Constant probability for each observation ex. Probability of getting a tail is the same each time we toss the coin
6 The Binomial Distribution: Properties Observations are independent The outcome of one observation does not affect the outcome of the other Two sampling methods Infinite population without replacement Finite population with replacement
7 Binomial Probabilities (Introduction) Suppose we toss a single fair and balanced coin five times in succession, and let X represent the number of heads. There are 5 = 3 possible sequences of H and T (S and F) in the sample space for this experiment. Of these, there are 10 in which there are exactly heads (X=): HHTTT HTHTT HTTHT HTTTH THHTT THTHT THTTH TTHHT TTHTH TTTHH The probability of each of these 10 outcomes is p q 3 = (1/) (1/) 3 =(1/3), so the probability of heads in 5 tosses of a fair and balanced coin is: P(X = ) = 10 * (1/3) = (10/3) = (1/3) Number of outcomes with heads Probability of each outcome with heads
8 Binomial Probabilities (continued) P(X=) = 10 * (1/3) = (10/3) =.315 Notice that this probability has two parts: 10 (1/3) Number of outcomes with heads Probability of each outcome with heads In general: 1. The probability of a given sequence of x successes out of n trials with probability of success p and probability of failure q is equal to: p x q (n-x). The number of different sequences of n trials that result in exactly x successes is equal to the number of choices of x elements out of a total of n elements. This number is denoted: n ncx x n! x!( n x)!
9 The Binomial Probability Distribution The binomial probability distribution: Px () n x pq n! x!( n x)! pq x ( nx) x ( nx) where : p is the probability of success in a single trial, q = 1-p, n is the number of trials, and x is the number of successes. N umber of successes, x Probability P(x) n! n 0 p q 0!( n 0)! n! 1 ( n 1) 1 p q 1!( n 1)! n! n p q!( n )! n! n 3 p q 3!( n 3)! n! n p q n!( n n)! 0 ( 0) ( ) 3 ( 3) n ( n n) 1.00
10 The Cumulative Binomial Probability Table n=5 p x Cumulative Binomial Probability Distribution and Binomial Probability Distribution of H,the Number of Heads Appearing in Five Tosses of a Fair Coin h F(h) P(h) Deriving Individual Probabilities from Cumulative Probabilities Fx ( ) PX ( x) Pi ( ) P() 3 F() 3 F() all ix P(X) = F(x) - F(x - 1) For example:
11 Calculating Binomial Probabilities - Example 60% of of Brooke shares are are owned by by LeBow. A random sample of of shares is is chosen. What is is the the probability that at at most three of of them will be be found to to be be owned by by LeBow? Check by by excel if if table values are are not available. n=15 p F ( x) F (3) P( X P( X x) alli x 3) 0.00 P( i)
12 Mean, Variance, and Standard Deviation of the Binomial Distribution Mean of a binomial distribution: E( X) np Variance of a binomial distribution: V ( X ) npq Standard deviation of a binomial distribution: For example, if H counts the number of heads in five tosses of a fair coin : E( H ) (5)(.5).5 H V ( H ) (5)(.5)(.5) 1.5 H = SD(X) = npq H SD( H )
13 Shape of the Binomial Distribution p = 0.1 p = 0.3 p = 0.5 Binomial Probability: n=4 p=0.1 Binomial Probability: n=4 p=0.3 Binomial Probability: n=4 p= n = 4 P(x) P(x) P(x) x x x 3 4 Binomial Probability: n=10 p=0.1 Binomial Probability: n=10 p=0.3 Binomial Probabil i ty: n=10 p= n = 10 P(x) P(x) P(x) x x x Binomial Probability: n=0 p=0.1 Binomial Probability: n=0 p=0.3 Binomial Probability: n=0 p=0.5 n = 0 P(x) P(x) P(x) x x x Binomial distributions become more symmetric as n increases and as p 0.5.
14 Example What is the probability of one success in five observations if the probability of success is.1? X = 1, n = 5, and p =.1 P(X 1) n! p X!(n X)! 5! (.1) 1!(5 1)! X 1 (1 p) (1.1) nx 51 (5)(.1)(.9)
15 Example Suppose the probability of purchasing a defective computer is 0.0. What is the probability of purchasing defective computers is a lot of 10?
16 X =, n = 10, and p =.0 P(X ) n! X!(n X)! p X (1 p) nx 10!!(10 (.0) )! (1.0) 10 (45)(.0004)(.8508).01531
17 Problem A manufacturing company of south Maharashtra found that after launching a golden handshake scheme for voluntary retirement, 10% of workers are unemployed. What is the probability of obtaining three or fewer unemployed workers in a random sample of 30 in a survey conducted by the company?
18 We have to find out the probability of getting (a) zero unemployed, x = 0; (b) one unemployed, x = 1; (c) two unemployed x = ; and (d) three unemployed, x = 3 workers.
19 Problem According to the U.S. Census Bureau, approximately 6% of all workers in jackson, Mississippi, are unemployed. In conducting a random telephone survey in Jackson, what is the probability of getting two or fewer unemployed workers in a sample of 0?
20 Answer n 0 p. 06 q. 94 PX ( ) PX ( 0) PX ( 1) PX ( ) P( X 0) 0! 0!(0 0)! ()()( ). 901 P( X 1) 0! 1!( 0 1)! ( 0)(. 06)(. 3086) P( X ) 0!!( 0 )! ( 190)(. 0036)(. 383). 46
21 The Binomial Distribution Using Binomial Tables n = 10 x p=.0 p=.5 p=.30 p=.35 p=.40 p=.45 p= Examples: p=.80 p=.75 p=.70 p=.65 p=.60 p=.55 p=.50 x n = 10, p =.35, x = 3: P(x = 3 n =10, p =.35) =.5 n = 10, p =.75, x = : P(x = n =10, p =.75) =.0004
22 Poisson Distribution: Applications Arrivals at queuing systems airports -- people, airplanes, automobiles, baggage banks -- people, automobiles, loan applications computer file servers -- read and write operations The number of scratches in a car s paint The number of mosquito bites on a person The number of computer crashes in a day Defects in manufactured goods number of defects per 1,000 feet of extruded copper wire number of blemishes per square foot of painted surface number of errors per typed page
23 The Poisson Distribution The Poisson probability distribution is useful for determining the probability of a number of occurrences over a given period of time or within a given area or volume. That is, the Poisson random variable counts occurrences over a continuous interval of time or space. It can also be used to calculate approximate binomial probabilities when the probability of success is small (p 0.05) and the number of trials is large (n 0). Poisson Distribution: x e P( x) for x = 1,,3,... x! where is the mean of the distribution (which also happens to be the variance) and e is the base of natural logarithms (e= ).
24 The Poisson Distribution Properties Apply the Poisson Distribution when: You wish to count the number of times an event occurs in a given area of opportunity The probability that an event occurs in one area of opportunity is the same for all areas of opportunity The number of events that occur in one area of opportunity is independent of the number of events that occur in the other areas of opportunity The probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller The average number of events per unit is (lambda)
25 Example Suppose that, on average, 5 cars enter a parking lot per minute. What is the probability that in a given minute, 7 cars will enter? So, X = 7 and λ = 5 P(7) λ e λ X! x e 5 5 7! So, there is a 10.4% chance 7 cars will enter the parking in a given minute.
26 Bank customer arrive randomly on weekday afternoons at an average of 3. customer every 4 minutes. A. What is the probability of having 10 customers every 8 minutes. B. What is the probability of having 6 customers every 8 minutes. C. What is the probability of having more than 7 customers in a 4-minute interval on a weekday afternoon. 3. customers / 4 minutes X = 10 customers / 8 minutes Adjusted = 64. customers / 8 minutes P(X) = X e X! PX ( = 10)= e 10! customers / 4 minutes X = 6 customers / 8 minutes Adjusted = 64. customers / 8 minutes P(X) = X e PX ( = 6)= X! e 6!
27 Poisson Distribution: Probability Table X
28 Poisson Distribution: Using the Poisson Tables X PX ( 4)
29 Poisson Distribution: Using the Poisson Tables X PX ( 5) PX ( 6) PX ( 7) PX ( 8) PX ( 9)
30 Poisson Distribution: Using the Poisson Tables X PX ( ) 1PX ( ) 1PX ( 0) PX ( 1)
31 Poisson Approximation of the Binomial Distribution Binomial probabilities are difficult to calculate when n is large. Under certain conditions binomial probabilities may be approximated by Poisson probabilities. If n 0 and n p 7, the approximation is acceptable. Use n p. Poisson approximation
32
33 The Poisson Distribution - Example Example Telephone manufacturers now offer 1000 different choices for a telephone (as combinations of color, type, options, portability, etc.). A company is opening a large regional office, and each of its 00 managers is allowed to order his or her own choice of a telephone. Assuming independence of choices and that each of the 1000 choices is equally likely, what is the probability that a particular choice will be made by none, one, two, or three of the managers? n = 00 = np = (00)(0.001) = 0. p = 1/1000 = e P( 0) 0!. e P() 1 1!. e P( )!. e P() 3 3! = = = =
34 The Poisson Distribution (continued) =1.0 = P(x) 0. P(x) X X =4 = P(x) 0.1 P(x) X X
35 The Poisson Distribution Using Poisson Tables X Example: Find P(X = ) if =.50 P(X ) λ e λ X! X e 0.50 (0.50)!.0758
36 The Poisson Distribution Shape 0.70 = X 0 P(X) P(x) P(X = ) =.0758 x
37 The Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameter : 0.70 = 0.50 = P(x) P(x) x x
38 Poisson Approximation of the Binomial Distribution Binomial X Poisson 15. n 50 p. 03 Error X Poisson 30. Binomial n 10, 000 p Error
39 Discrete and Continuous Random Variables - Revisited A discrete random variable: counts occurrences has a countable number of possible values has discrete jumps between successive values has measurable probability associated with individual values probability is height A continuous random variable: measures (e.g.: height, weight, speed, value, duration, length) has an uncountably infinite number of possible values moves continuously from value to value has no measurable probability associated with individual values probability is area For example: Binomial n=3 p=.5 x P(x) P(x) Binomial: n=3 p=.5 1 C1 3 For example: In this case, the shaded area epresents the probability that the task takes between and 3 minutes. P(x) MinutestoCompleteTask Minutes
40 From a Discrete to a Continuous Distribution The time it takes to complete a task can be subdivided into: Half-Minute Intervals Quarter-Minute Intervals Eighth-Minute Intervals 0.15 Minutes to Complete Task: By Half-Minutes MinutestoCompleteTask:FourthsofaMinute Minutes to Complete Task: Eighths of aminute 0.10 P(x) P(x) P(x) Minutes Minutes Minutes Or even infinitesimally small intervals: Minutes to Complete Task: Probability Density Function f(z) When When a a continuous continuous random random variable variable has has been been subdivided subdivided into into infinitesimally infinitesimally small small intervals, intervals, a a measurable measurable probability probability can can only only be be associated associated with with an an interval interval of of values, values, and and the the probability probability is is given given by by the the area area beneath beneath the the probability probability density density function function corresponding corresponding to to that that interval. interval. In In this this example, example, the the shaded shaded area area represents represents P( P( X ). ) Minutes
41 Continuous Random Variables A continuous random variable is is a a random random variable variable that that can can take take on on any any value value in in an an interval interval of of numbers. The The probabilities associated with with a a continuous random random variable variable X are are determined by by the the probability density density function of of the the random random variable. The The function, denoted denoted f(x), f(x), has has the the following properties f(x) f(x) 0 for for all all x. x... The The probability that that X will will be be between two two numbers a and and b is is equal equal to to the the area area under under f(x) f(x) between a and and b. b The The total total area area under under the the curve curve of of f(x) f(x) is is equal equal to to The The cumulative distribution function of of a a continuous random random variable: F(x) F(x) = P(X P(X x) x) =Area =Area under under f(x) f(x) between between the the smallest smallest possible possible value value of of X (often (often-) -) and and the the point point x. x.
42 Probability Density Function and Cumulative Distribution Function F(x) 1 F(b) F(a) } P(a X b)=f(b) - F(a) f(x) 0 a b x P(a X b) = Area under f(x) between a and b = F(b) - F(a) 0 a b x
Engineering Mathematics III
The Binomial, Poisson, and Normal Distributions Probability distributions We use probability distributions because they work they fit lots of data in real world 100 80 60 40 20 Std. Dev = 14.76 Mean =
More informationThe Binomial, Poisson, and Normal Distributions. Engineering Mathematics III
The Binomial, Poisson, and Normal Distributions Probability distributions We use probability distributions because they work they fit lots of data in real world 8 6 4 Std. Dev = 4.76 Mean = 35.3 N = 73..
More informationChapter 3 Probability Distribution
Chapter 3 Probability Distribution Probability Distributions A probability function is a function which assigns probabilities to the values of a random variable. Individual probability values may be denoted
More informationStatistics for Managers Using Microsoft Excel (3 rd Edition)
Statistics for Managers Using Microsoft Excel (3 rd Edition) Chapter 4 Basic Probability and Discrete Probability Distributions 2002 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic probability concepts
More informationBernoulli Trials, Binomial and Cumulative Distributions
Bernoulli Trials, Binomial and Cumulative Distributions Sec 4.4-4.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 9-3339 Cathy Poliak,
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1 Chapter Topics Basic Probability Concepts: Sample
More informationDistribusi Binomial, Poisson, dan Hipergeometrik
Distribusi Binomial, Poisson, dan Hipergeometrik CHAPTER TOPICS The Probability of a Discrete Random Variable Covariance and Its Applications in Finance Binomial Distribution Poisson Distribution Hypergeometric
More informationBinomial and Poisson Probability Distributions
Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016 Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What
More informationChapter 2: Discrete Distributions. 2.1 Random Variables of the Discrete Type
Chapter 2: Discrete Distributions 2.1 Random Variables of the Discrete Type 2.2 Mathematical Expectation 2.3 Special Mathematical Expectations 2.4 Binomial Distribution 2.5 Negative Binomial Distribution
More informationSuppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8.
Suppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8. Coin A is flipped until a head appears, then coin B is flipped until
More informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 06 McGraw-Hill/Irwin Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved. LEARNING OBJECTIVES LO 6-1 Identify the characteristics of a probability
More informationKnown probability distributions
Known probability distributions Engineers frequently wor with data that can be modeled as one of several nown probability distributions. Being able to model the data allows us to: model real systems design
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationRandom Variable And Probability Distribution. Is defined as a real valued function defined on the sample space S. We denote it as X, Y, Z,
Random Variable And Probability Distribution Introduction Random Variable ( r.v. ) Is defined as a real valued function defined on the sample space S. We denote it as X, Y, Z, T, and denote the assumed
More informationProbability and Statistics for Engineers
Probability and Statistics for Engineers Chapter 4 Probability Distributions Ruochen Liu Ruochenliu@xidian.edu.cn Institute of Intelligent Information Processing, Xidian University Outline Random variables
More informationUnit II. Page 1 of 12
Unit II (1) Basic Terminology: (i) Exhaustive Events: A set of events is said to be exhaustive, if it includes all the possible events. For example, in tossing a coin there are two exhaustive cases either
More informationDiscrete Distributions
A simplest example of random experiment is a coin-tossing, formally called Bernoulli trial. It happens to be the case that many useful distributions are built upon this simplest form of experiment, whose
More informationRandom Variable. Discrete Random Variable. Continuous Random Variable. Discrete Random Variable. Discrete Probability Distribution
Random Variable Theoretical Probability Distribution Random Variable Discrete Probability Distributions A variable that assumes a numerical description for the outcome of a random eperiment (by chance).
More informationIntroduction to Probability and Statistics Slides 3 Chapter 3
Introduction to Probability and Statistics Slides 3 Chapter 3 Ammar M. Sarhan, asarhan@mathstat.dal.ca Department of Mathematics and Statistics, Dalhousie University Fall Semester 2008 Dr. Ammar M. Sarhan
More informationTopic 3 - Discrete distributions
Topic 3 - Discrete distributions Basics of discrete distributions Mean and variance of a discrete distribution Binomial distribution Poisson distribution and process 1 A random variable is a function which
More informationWhat is Probability? Probability. Sample Spaces and Events. Simple Event
What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5
More informationBinomial random variable
Binomial random variable Toss a coin with prob p of Heads n times X: # Heads in n tosses X is a Binomial random variable with parameter n,p. X is Bin(n, p) An X that counts the number of successes in many
More informationSTAT509: Discrete Random Variable
University of South Carolina September 16, 2014 Motivation So far, we have already known how to calculate probabilities of events. Suppose we toss a fair coin three times, we know that the probability
More informationIntroduction to Statistical Data Analysis Lecture 3: Probability Distributions
Introduction to Statistical Data Analysis Lecture 3: Probability Distributions James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis
More informationAn-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)
An-Najah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 3 Discrete Random
More informationS2 QUESTIONS TAKEN FROM JANUARY 2006, JANUARY 2007, JANUARY 2008, JANUARY 2009
S2 QUESTIONS TAKEN FROM JANUARY 2006, JANUARY 2007, JANUARY 2008, JANUARY 2009 SECTION 1 The binomial and Poisson distributions. Students will be expected to use these distributions to model a real-world
More informationDiscrete Random Variables. Discrete Random Variables
Random Variables In many situations, we are interested in numbers associated with the outcomes of a random experiment. For example: Testing cars from a production line, we are interested in variables such
More informationMATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3
MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3 1. A four engine plane can fly if at least two engines work. a) If the engines operate independently and each malfunctions with probability q, what is the
More informationChapter 1: Revie of Calculus and Probability
Chapter 1: Revie of Calculus and Probability Refer to Text Book: Operations Research: Applications and Algorithms By Wayne L. Winston,Ch. 12 Operations Research: An Introduction By Hamdi Taha, Ch. 12 OR441-Dr.Khalid
More informationsuccess and failure independent from one trial to the next?
, section 8.4 The Binomial Distribution Notes by Tim Pilachowski Definition of Bernoulli trials which make up a binomial experiment: The number of trials in an experiment is fixed. There are exactly two
More informationREPEATED TRIALS. p(e 1 ) p(e 2 )... p(e k )
REPEATED TRIALS We first note a basic fact about probability and counting. Suppose E 1 and E 2 are independent events. For example, you could think of E 1 as the event of tossing two dice and getting a
More informationApplied Statistics I
Applied Statistics I (IMT224β/AMT224β) Department of Mathematics University of Ruhuna A.W.L. Pubudu Thilan Department of Mathematics University of Ruhuna Applied Statistics I(IMT224β/AMT224β) 1/158 Chapter
More informationLecture 4 : The Binomial Distribution. Jonathan Marchini
Lecture 4 : The Binomial Distribution Jonathan Marchini Permutations and Combinations (Recap) Consider 7 students applying to a college for 3 places Abi Ben Claire Dave Emma Frank Gail How many ways are
More informationSystem Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models
System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 20, 2012 Introduction Introduction The world of the model-builder
More informationChapter 2: The Random Variable
Chapter : The Random Variable The outcome of a random eperiment need not be a number, for eample tossing a coin or selecting a color ball from a bo. However we are usually interested not in the outcome
More informationWeek 6, 9/24/12-9/28/12, Notes: Bernoulli, Binomial, Hypergeometric, and Poisson Random Variables
Week 6, 9/24/12-9/28/12, Notes: Bernoulli, Binomial, Hypergeometric, and Poisson Random Variables 1 Monday 9/24/12 on Bernoulli and Binomial R.V.s We are now discussing discrete random variables that have
More informationDiscrete probability distributions
Discrete probability s BSAD 30 Dave Novak Fall 08 Source: Anderson et al., 05 Quantitative Methods for Business th edition some slides are directly from J. Loucks 03 Cengage Learning Covered so far Chapter
More informationRelationship between probability set function and random variable - 2 -
2.0 Random Variables A rat is selected at random from a cage and its sex is determined. The set of possible outcomes is female and male. Thus outcome space is S = {female, male} = {F, M}. If we let X be
More informationChapter 7: Theoretical Probability Distributions Variable - Measured/Categorized characteristic
BSTT523: Pagano & Gavreau, Chapter 7 1 Chapter 7: Theoretical Probability Distributions Variable - Measured/Categorized characteristic Random Variable (R.V.) X Assumes values (x) by chance Discrete R.V.
More informationGeometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last
Geometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating
More informationBinomial Distribution. Collin Phillips
Mathematics Learning Centre Binomial Distribution Collin Phillips c 00 University of Sydney Thanks To Darren Graham and Cathy Kennedy for turning my scribble into a book and to Jackie Nicholas and Sue
More informationCHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS
CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS 4.2 Events and Sample Space De nition 1. An experiment is the process by which an observation (or measurement) is obtained Examples 1. 1: Tossing a pair
More informationStatistical Experiment A statistical experiment is any process by which measurements are obtained.
(التوزيعات الا حتمالية ( Distributions Probability Statistical Experiment A statistical experiment is any process by which measurements are obtained. Examples of Statistical Experiments Counting the number
More informationMgtOp 215 Chapter 5 Dr. Ahn
MgtOp 215 Chapter 5 Dr. Ahn Random variable: a variable that assumes its values corresponding to a various outcomes of a random experiment, therefore its value cannot be predicted with certainty. Discrete
More informationProbability Theory. Introduction to Probability Theory. Principles of Counting Examples. Principles of Counting. Probability spaces.
Probability Theory To start out the course, we need to know something about statistics and probability Introduction to Probability Theory L645 Advanced NLP Autumn 2009 This is only an introduction; for
More informationStatistics for Economists. Lectures 3 & 4
Statistics for Economists Lectures 3 & 4 Asrat Temesgen Stockholm University 1 CHAPTER 2- Discrete Distributions 2.1. Random variables of the Discrete Type Definition 2.1.1: Given a random experiment with
More informationChapter 3. Discrete Random Variables and Their Probability Distributions
Chapter 3. Discrete Random Variables and Their Probability Distributions 2.11 Definition of random variable 3.1 Definition of a discrete random variable 3.2 Probability distribution of a discrete random
More informationDiscrete Random Variable Practice
IB Math High Level Year Discrete Probability Distributions - MarkScheme Discrete Random Variable Practice. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The
More informationSTA 584 Supplementary Examples (not to be graded) Fall, 2003
Page 1 of 8 Central Michigan University Department of Mathematics STA 584 Supplementary Examples (not to be graded) Fall, 003 1. (a) If A and B are independent events, P(A) =.40 and P(B) =.70, find (i)
More informationBus 216: Business Statistics II Introduction Business statistics II is purely inferential or applied statistics.
Bus 216: Business Statistics II Introduction Business statistics II is purely inferential or applied statistics. Study Session 1 1. Random Variable A random variable is a variable that assumes numerical
More informationQuantitative Methods for Decision Making
January 14, 2012 Lecture 3 Probability Theory Definition Mutually exclusive events: Two events A and B are mutually exclusive if A B = φ Definition Special Addition Rule: Let A and B be two mutually exclusive
More informationTOPIC 12 PROBABILITY SCHEMATIC DIAGRAM
TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos
More informationFind the value of n in order for the player to get an expected return of 9 counters per roll.
. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters
More informationPROBABILITY.
PROBABILITY PROBABILITY(Basic Terminology) Random Experiment: If in each trial of an experiment conducted under identical conditions, the outcome is not unique, but may be any one of the possible outcomes,
More informationII. The Binomial Distribution
88 CHAPTER 4 PROBABILITY DISTRIBUTIONS 進佳數學團隊 Dr. Herbert Lam 林康榮博士 HKDSE Mathematics M1 II. The Binomial Distribution 1. Bernoulli distribution A Bernoulli eperiment results in any one of two possible
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER / Probability
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER 2 2017/2018 DR. ANTHONY BROWN 5.1. Introduction to Probability. 5. Probability You are probably familiar with the elementary
More information(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationThere are two basic kinds of random variables continuous and discrete.
Summary of Lectures 5 and 6 Random Variables The random variable is usually represented by an upper case letter, say X. A measured value of the random variable is denoted by the corresponding lower case
More informationDISCRETE VARIABLE PROBLEMS ONLY
DISCRETE VARIABLE PROBLEMS ONLY. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The table below shows the possible scores on the die, the probability of each
More informationNotes for Math 324, Part 17
126 Notes for Math 324, Part 17 Chapter 17 Common discrete distributions 17.1 Binomial Consider an experiment consisting by a series of trials. The only possible outcomes of the trials are success and
More informationDiscrete Probability Distribution
Shapes of binomial distributions Discrete Probability Distribution Week 11 For this activity you will use a web applet. Go to http://socr.stat.ucla.edu/htmls/socr_eperiments.html and choose Binomial coin
More informationProbability and Statistics Concepts
University of Central Florida Computer Science Division COT 5611 - Operating Systems. Spring 014 - dcm Probability and Statistics Concepts Random Variable: a rule that assigns a numerical value to each
More informationEach trial has only two possible outcomes success and failure. The possible outcomes are exactly the same for each trial.
Section 8.6: Bernoulli Experiments and Binomial Distribution We have already learned how to solve problems such as if a person randomly guesses the answers to 10 multiple choice questions, what is the
More informationEDEXCEL S2 PAPERS MARK SCHEMES AVAILABLE AT:
EDEXCEL S2 PAPERS 2009-2007. MARK SCHEMES AVAILABLE AT: http://www.physicsandmathstutor.com/a-level-maths-papers/s2-edexcel/ JUNE 2009 1. A bag contains a large number of counters of which 15% are coloured
More informationChapter 3: Discrete Random Variable
Chapter 3: Discrete Random Variable Shiwen Shen University of South Carolina 2017 Summer 1 / 63 Random Variable Definition: A random variable is a function from a sample space S into the real numbers.
More informationELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random
More informationLecture 16. Lectures 1-15 Review
18.440: Lecture 16 Lectures 1-15 Review Scott Sheffield MIT 1 Outline Counting tricks and basic principles of probability Discrete random variables 2 Outline Counting tricks and basic principles of probability
More informationProbability Distribution
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationBINOMIAL DISTRIBUTION
BINOMIAL DISTRIBUTION The binomial distribution is a particular type of discrete pmf. It describes random variables which satisfy the following conditions: 1 You perform n identical experiments (called
More informationMTH4451Test#2-Solutions Spring 2009
Pat Rossi Instructions. MTH4451Test#2-Solutions Spring 2009 Name Show CLEARLY how you arrive at your answers. 1. A large jar contains US coins. In this jar, there are 350 pennies ($0.01), 300 nickels ($0.05),
More informationQueueing Theory and Simulation. Introduction
Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University, Japan
More informationBernoulli Trials and Binomial Distribution
Bernoulli Trials and Binomial Distribution Sec 4.4-4.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 10-3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationDiscrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Introduction The markets can be thought of as a complex interaction of a large number of random
More informationLecture 6. Probability events. Definition 1. The sample space, S, of a. probability experiment is the collection of all
Lecture 6 1 Lecture 6 Probability events Definition 1. The sample space, S, of a probability experiment is the collection of all possible outcomes of an experiment. One such outcome is called a simple
More information14.2 THREE IMPORTANT DISCRETE PROBABILITY MODELS
14.2 THREE IMPORTANT DISCRETE PROBABILITY MODELS In Section 14.1 the idea of a discrete probability model was introduced. In the examples of that section the probability of each basic outcome of the experiment
More informationMTH302 Quiz # 4. Solved By When a coin is tossed once, the probability of getting head is. Select correct option:
MTH302 Quiz # 4 Solved By konenuchiha@gmail.com When a coin is tossed once, the probability of getting head is. 0.55 0.52 0.50 (1/2) 0.51 Suppose the slope of regression line is 20 and the intercept is
More informationReview of Probability. CS1538: Introduction to Simulations
Review of Probability CS1538: Introduction to Simulations Probability and Statistics in Simulation Why do we need probability and statistics in simulation? Needed to validate the simulation model Needed
More informationCOVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS
COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS COURSE: CBS 221 DISCLAIMER The contents of this document are intended for practice and leaning purposes at the undergraduate
More informationRandom Variables Example:
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More information(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationDept. of Linguistics, Indiana University Fall 2015
L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would
More informationFault-Tolerant Computer System Design ECE 60872/CS 590. Topic 2: Discrete Distributions
Fault-Tolerant Computer System Design ECE 60872/CS 590 Topic 2: Discrete Distributions Saurabh Bagchi ECE/CS Purdue University Outline Basic probability Conditional probability Independence of events Series-parallel
More informationSolutions - Final Exam
Solutions - Final Exam Instructors: Dr. A. Grine and Dr. A. Ben Ghorbal Sections: 170, 171, 172, 173 Total Marks Exercise 1 7 Exercise 2 6 Exercise 3 6 Exercise 4 6 Exercise 5 6 Exercise 6 9 Total 40 Score
More informationName: Firas Rassoul-Agha
Midterm 1 - Math 5010 - Spring 016 Name: Firas Rassoul-Agha Solve the following 4 problems. You have to clearly explain your solution. The answer carries no points. Only the work does. CALCULATORS ARE
More informationWeek 12-13: Discrete Probability
Week 12-13: Discrete Probability November 21, 2018 1 Probability Space There are many problems about chances or possibilities, called probability in mathematics. When we roll two dice there are possible
More informationTo find the median, find the 40 th quartile and the 70 th quartile (which are easily found at y=1 and y=2, respectively). Then we interpolate:
Joel Anderson ST 37-002 Lecture Summary for 2/5/20 Homework 0 First, the definition of a probability mass function p(x) and a cumulative distribution function F(x) is reviewed: Graphically, the drawings
More informationChapter 1 (Basic Probability)
Chapter 1 (Basic Probability) What is probability? Consider the following experiments: 1. Count the number of arrival requests to a web server in a day. 2. Determine the execution time of a program. 3.
More informationB.N.Bandodkar College of Science, Thane. Subject : Computer Simulation and Modeling.
B.N.Bandodkar College of Science, Thane Subject : Computer Simulation and Modeling. Simulation is a powerful technique for solving a wide variety of problems. To simulate is to copy the behaviors of a
More informationExpectations. Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or
Expectations Expectations Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or µ X, is E(X ) = µ X = x D x p(x) Expectations
More informationLecture 10. Variance and standard deviation
18.440: Lecture 10 Variance and standard deviation Scott Sheffield MIT 1 Outline Defining variance Examples Properties Decomposition trick 2 Outline Defining variance Examples Properties Decomposition
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationConditional Probability
Conditional Probability Conditional Probability The Law of Total Probability Let A 1, A 2,..., A k be mutually exclusive and exhaustive events. Then for any other event B, P(B) = P(B A 1 ) P(A 1 ) + P(B
More informationDiscrete and continuous
Discrete and continuous A curve, or a function, or a range of values of a variable, is discrete if it has gaps in it - it jumps from one value to another. In practice in S2 discrete variables are variables
More informationChapter 4 : Discrete Random Variables
STAT/MATH 394 A - PROBABILITY I UW Autumn Quarter 2015 Néhémy Lim Chapter 4 : Discrete Random Variables 1 Random variables Objectives of this section. To learn the formal definition of a random variable.
More informationDEFINITION: IF AN OUTCOME OF A RANDOM EXPERIMENT IS CONVERTED TO A SINGLE (RANDOM) NUMBER (E.G. THE TOTAL
CHAPTER 5: RANDOM VARIABLES, BINOMIAL AND POISSON DISTRIBUTIONS DEFINITION: IF AN OUTCOME OF A RANDOM EXPERIMENT IS CONVERTED TO A SINGLE (RANDOM) NUMBER (E.G. THE TOTAL NUMBER OF DOTS WHEN ROLLING TWO
More informationIEOR 3106: Introduction to Operations Research: Stochastic Models. Professor Whitt. SOLUTIONS to Homework Assignment 1
IEOR 3106: Introduction to Operations Research: Stochastic Models Professor Whitt SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability
More informationTopic 3: The Expectation of a Random Variable
Topic 3: The Expectation of a Random Variable Course 003, 2017 Page 0 Expectation of a discrete random variable Definition (Expectation of a discrete r.v.): The expected value (also called the expectation
More informationProbability Theory and Random Variables
Probability Theory and Random Variables One of the most noticeable aspects of many computer science related phenomena is the lack of certainty. When a job is submitted to a batch oriented computer system,
More informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 McGraw-Hill/Irwin Copyright 2012 by The McGraw-Hill Companies, Inc. All rights reserved. LO5 Describe and compute probabilities for a binomial distribution.
More informationLecture 4: Random Variables and Distributions
Lecture 4: Random Variables and Distributions Goals Random Variables Overview of discrete and continuous distributions important in genetics/genomics Working with distributions in R Random Variables A
More information