Random Variable. Discrete Random Variable. Continuous Random Variable. Discrete Random Variable. Discrete Probability Distribution
|
|
- Brendan Lang
- 6 years ago
- Views:
Transcription
1 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). Usually is denoted by a capital letter. X, Y, Z,... Very useful for mathematically modeling the distribution of variables. 1 2 Continuous Random Variable : A random variable assumes discrete values by chance. Continuous Random Variable: A random variable that can take on any value within a specified interval by chance. 3 4 Eample: (Toss a balanced coin) X = 1, if Head occurs, and X = 0, if Tail occurs. Head) = X=1) = 1) =.5 Tail) = X=0) = 0) =.5 1/2 Probability mass function: X=) =.5, if = 0, 1, 0 1 and X=) = 0, elsewhere. Total probability is 1. 5 Discrete Probability Distribution If a balanced coin is tossed, Head and Tail are equally likely to occur, X=1) =.5 = 1/2 and X=0) =.5 = 1/2 all possible outcomes) = X=1 or 0) = X=1) + X=0) = 1/2 + 1/2 = 1.0 Total probability is 1. 6 Random Variable - 1
2 Eample: What is probability of getting a number less than 3 when roll a balanced die? Probability mass function: X=) =1/6, if =1,2,3,4,5,or 6, and X=) = 0 elsewhere. 1/ X < 3 ) = X 2) = X = 1) + X = 2) = 1/6 + 1/6 = 2/6 7 Discrete Uniform Probability Distribution Probability Mass Function for Discrete Uniform Distribution: ) = c, c is a constant Balanced Coin: ) = 1/2, for = 0,1 Balanced Die: ) = 1/6, for = 1,2,3,4,5,6 8 Balanced Die: ) = 1/6, for = 1,2,3,4,5,6 X=3) = 1/6 X=5) = 1/6 Other Discrete Distributions Bernoulli Binomial Poisson Geometric 9 10 Bernoulli Trial Bernoulli Distribution Model (Bernoulli Probability Distribution) Definition: Bernoulli trial is a random eperiment whose outcomes are classified as one of the two categories. (S, F) or (Success, Failure) or (1, 0) Eample: Tossing a coin, observing Head or Tail Observing patient s status Died or Survived Random Variable - 2
3 Bernoulli Probability Distribution Eample: (Tossing a balanced coin) S) = X=1) = p =.5 F) = X=0) = 1 p =.5 Bernoulli Distribution Bernoulli Probability Distribution Eample: In a random eperiment of casting a balanced die, we are only interested in observing 6 turns up or not. It is a Bernoulli trail. 6) = X=1) = p = 1/6 6 ) = X=0) =1 1/6 = 5/ Bernoulli Distribution Binomial Eperiment Binomial Distribution Model (Binomial Probability Distribution) 15 A random eperiment involving a sequence of independent and identical Bernoulli trials. Eample: Toss a coin ten times, and observing Head turns up. Roll a die 3 times, and observing a 6 turns up or not. In a random sample of 5 from a large population, and observing subjects disease status. (Almost binomial) 16 Binomial Probability Model A model to find the probability of having number successes in a sequence of n independent and identical Bernoulli trials. 17 Binomial Probability Model In a binomial eperiment involving n independent and identical Bernoulli trials each with probability of success p, the probability of having successes can be calculated with the binomial probability mass function, and it is, for = 0, 1,, n, n! n X = ) = p (1!( n )! n = p (1 n 18 Random Variable - 3
4 Factorial n! = n 0! = 1 Eample: 3! = = 6 5 5! Eample: = = = 10 2 (5 2)!2! Binomial Probability Eample: A balanced die is rolled three times (or three balanced dice are rolled), what is the probability to see two 6 s? Identify n = 3, p = 1/6, = 2 (6, 6, 6 ) (6, 6, 6) (6, 6, 6 ) 3! X=2) = (1/6) 2 (5/6) 3-2 2! 1! = 3 (1/6) 2 (5/6) 1 =.069 n! n X = ) = p (1!( n )! 20 Binomial Probability Binomial Probability Eample: If there are 10% of the population in a community have a certain disease, what is the probability that 4 people in a random sample of 5 people from this community has the disease? (Assume binomial eperiment.) Identify n = 5, = 4, p =.10 5! X=4) = (.10) 4 (1.10) 5-4 4! 1! = 5 (.10) 4 (.90) 1 =.0004 n! n X = ) = p (1!( n )! 21 Eample: In the previous problem, what is the probability that 4 or more people have the disease? Identify n = 5, = 4, p =.10 X 4) = X=4) + X=5) 5! = (.10) 5 (1.10) 5-5 5! 0! = = (What is this number telling us?) 22 Parameters of Binomial Distribution Parameters of the distribution: Mean of the distribution, µ =n p Variance of the distribution, σ 2 = n p (1 Standard deviation, σ, is the square root of variance. Binomial Distribution n = 5, p =.10 µ = 5.10 =.5 σ 2 = 5.1 (1.1) = ) = ) = ) = ) = ) = ) = Random Variable - 4
5 Discrete Probability Models Bernoulli : Two categories of outcomes. Binomial : Number of successes in a binomial eperiment. Poisson : Number of successes in a given time period or in a given unit space. 25 Poisson Distribution Let X represents the number of occurrences of some event of interest over a given interval from a Poisson process, and the λ is the mean of the distribution, then the probability of observing occurrences is, for = 0, 1, 2,, λ X = ) = e λ! It can be used to approimate Binomial prob., for large n. e = Poisson Process The probability that a single event occurs within an interval is proportional to the length of the interval. Within a single interval, an infinite number of occurrences is possible. The events occur independently both within the same interval and between consecutive non-overlapping intervals. Eamples of Poisson Process Number of people visiting to the emergency room for treatment per hour. Number of customers coming to the Arby s to buy sandwich per ten minutes Poisson Probability If on average there are 4 people catch flu in a given week in a community during a certain season, what is the probability of observing 2 people catch flu in this community in a given week period during the season? (Assume the number of people catching flu in a given period of time follow a Poisson Process.) Poisson Probability If on average there are 4 people catch flu in a given week in a community during a certain season, what is the probability of observing 2 people catch flu in this community in a given two weeks period during the season? (Assume the number of people catching flu in a given period of time follow a Poisson Process.) λ = 4 = 2 e X=2) = =.1465 X = ) = e λ λ! λ = 42 = 8 = 2 e X=2) = =.011 X = ) = e λ λ! 2! 2! Random Variable - 5
6 Poisson Probability If on average there are 4 people catch flu in a given week in a community during a certain season, what is the probability of observing less than 2 people catch flu in this community in a given week period during the season? (Assume the number of people catching flu in a given period of time follow a Poisson Process.) λ = 4 = 0 and 1 X<2) = X=0) + X=1) Discrete Probability Models Binomial : Number of successes in a binomial eperiment. (There is a sample taken.) Poisson : Number of successes in a given time period or in a given unit space. (No sample taken.) = (e )/0! + (e )/1! = Random Variable - 6
Binomial 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 informationConditional Probability
Conditional Probability Idea have performed a chance experiment but don t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what s the
More informationChapter 3 Single Random Variables and Probability Distributions (Part 1)
Chapter 3 Single Random Variables and Probability Distributions (Part 1) Contents What is a Random Variable? Probability Distribution Functions Cumulative Distribution Function Probability Density Function
More informationProbability Rules. MATH 130, Elements of Statistics I. J. Robert Buchanan. Fall Department of Mathematics
Probability Rules MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Introduction Probability is a measure of the likelihood of the occurrence of a certain behavior
More informationProbability Experiments, Trials, Outcomes, Sample Spaces Example 1 Example 2
Probability Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application. However, probability models underlie
More informationChapter 2. Random Variable. Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance.
Chapter 2 Random Variable CLO2 Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance. 1 1. Introduction In Chapter 1, we introduced the concept
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 information18440: Probability and Random variables Quiz 1 Friday, October 17th, 2014
18440: Probability and Random variables Quiz 1 Friday, October 17th, 014 You will have 55 minutes to complete this test. Each of the problems is worth 5 % of your exam grade. No calculators, notes, or
More informationPart 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park August 19, 2008 1 Introduction There are three main objectives to this section: 1. To introduce the concepts of probability and random variables.
More informationCS 361: Probability & Statistics
February 26, 2018 CS 361: Probability & Statistics Random variables The discrete uniform distribution If every value of a discrete random variable has the same probability, then its distribution is called
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 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 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 informationIntroductory Probability
Introductory Probability Bernoulli Trials and Binomial Probability Distributions Dr. Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK February 04, 2019 Agenda Bernoulli Trials and Probability
More informationThe random variable 1
The random variable 1 Contents 1. Definition 2. Distribution and density function 3. Specific random variables 4. Functions of one random variable 5. Mean and variance 2 The random variable A random variable
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 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 informationCMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 5 Prof. Hanna Wallach wallach@cs.umass.edu February 7, 2012 Reminders Pick up a copy of B&T Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
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 informationRandom variable X is a mapping that maps each outcome s in the sample space to a unique real number x, x. X s. Real Line
Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number,. s s : outcome Sample Space Real Line Eamples Toss a coin. Define the random variable
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 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 informationPOISSON RANDOM VARIABLES
POISSON RANDOM VARIABLES Suppose a random phenomenon occurs with a mean rate of occurrences or happenings per unit of time or length or area or volume, etc. Note: >. Eamples: 1. Cars passing through an
More informationRandom variable X is a mapping that maps each outcome s in the sample space to a unique real number x, < x <. ( ) X s. Real Line
Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number, <
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 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 informationProbability Density Functions and the Normal Distribution. Quantitative Understanding in Biology, 1.2
Probability Density Functions and the Normal Distribution Quantitative Understanding in Biology, 1.2 1. Discrete Probability Distributions 1.1. The Binomial Distribution Question: You ve decided to flip
More informationProbability Method in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institute of Technology, Kharagpur
Probability Method in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture No. # 34 Probability Models using Discrete Probability Distributions
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 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 informationMATH 10 INTRODUCTORY STATISTICS
MATH 10 INTRODUCTORY STATISTICS Ramesh Yapalparvi Week 2 Chapter 4 Bivariate Data Data with two/paired variables, Pearson correlation coefficient and its properties, general variance sum law Chapter 6
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 informationContinuous-Valued Probability Review
CS 6323 Continuous-Valued Probability Review Prof. Gregory Provan Department of Computer Science University College Cork 2 Overview Review of discrete distributions Continuous distributions 3 Discrete
More informationLecture 2. Binomial and Poisson Probability Distributions
Durkin, Lecture 2, Page of 6 Lecture 2 Binomial and Poisson Probability Distributions ) Bernoulli Distribution or Binomial Distribution: Consider a situation where there are only two possible outcomes
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 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 informationBusiness Statistics PROBABILITY DISTRIBUTIONS
Business Statistics PROBABILITY DISTRIBUTIONS CONTENTS Probability distribution functions (discrete) Characteristics of a discrete distribution Example: uniform (discrete) distribution Example: Bernoulli
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 informationSenior Math Circles November 19, 2008 Probability II
University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles November 9, 2008 Probability II Probability Counting There are many situations where
More informationSome Special Discrete Distributions
Mathematics Department De La Salle University Manila February 6, 2017 Some Discrete Distributions Often, the observations generated by different statistical experiments have the same general type of behaviour.
More informationCS 1538: Introduction to Simulation Homework 1
CS 1538: Introduction to Simulation Homework 1 1. A fair six-sided die is rolled three times. Let X be a random variable that represents the number of unique outcomes in the three tosses. For example,
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 informationPart 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park April 3, 2009 1 Introduction Is the coin fair or not? In part one of the course we introduced the idea of separating sampling variation from a
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 information4. Discrete Probability Distributions. Introduction & Binomial Distribution
4. Discrete Probability Distributions Introduction & Binomial Distribution Aim & Objectives 1 Aims u Introduce discrete probability distributions v Binomial distribution v Poisson distribution 2 Objectives
More informationLecture 2 Binomial and Poisson Probability Distributions
Binomial Probability Distribution Lecture 2 Binomial and Poisson Probability Distributions Consider a situation where there are only two possible outcomes (a Bernoulli trial) Example: flipping a coin James
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 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 informationSection 2.4 Bernoulli Trials
Section 2.4 Bernoulli Trials A bernoulli trial is a repeated experiment with the following properties: 1. There are two outcomes of each trial: success and failure. 2. The probability of success in each
More information1. If X has density. cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. f(x) =
1. If X has density f(x) = { cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. 2. Let X have density f(x) = { xe x, 0 < x < 0, otherwise. (a) Find P (X > 2). (b) Find
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 informationProbability: Terminology and Examples Class 2, Jeremy Orloff and Jonathan Bloom
1 Learning Goals Probability: Terminology and Examples Class 2, 18.05 Jeremy Orloff and Jonathan Bloom 1. Know the definitions of sample space, event and probability function. 2. Be able to organize a
More informationEngineering 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 informationTutorial 2: Probability
1 Tutorial 2: Probability 1. Coin Tosses Recall that the probability of a specified outcome or outcome(s) is given by prob. of success = number of successful outcomes total number of possible outcomes.
More informationUNIT NUMBER PROBABILITY 6 (Statistics for the binomial distribution) A.J.Hobson
JUST THE MATHS UNIT NUMBER 19.6 PROBABILITY 6 (Statistics for the binomial distribution) by A.J.Hobson 19.6.1 Construction of histograms 19.6.2 Mean and standard deviation of a binomial distribution 19.6.3
More informationSome Famous Discrete Distributions. 1. Uniform 2. Bernoulli 3. Binomial 4. Negative Binomial 5. Geometric 6. Hypergeometric 7.
Some Famous Discrete Distributions. Uniform 2. Bernoulli 3. Binomial 4. Negative Binomial 5. Geometric 6. Hypergeometric 7. Poisson 5.2: Discrete Uniform Distribution: If the discrete random variable X
More informationChapter (4) Discrete Probability Distributions Examples
Chapter (4) Discrete Probability Distributions Examples Example () Two balanced dice are rolled. Let X be the sum of the two dice. Obtain the probability distribution of X. Solution When the two balanced
More informationMath 493 Final Exam December 01
Math 493 Final Exam December 01 NAME: ID NUMBER: Return your blue book to my office or the Math Department office by Noon on Tuesday 11 th. On all parts after the first show enough work in your exam booklet
More informationLecture #13 Tuesday, October 4, 2016 Textbook: Sections 7.3, 7.4, 8.1, 8.2, 8.3
STATISTICS 200 Lecture #13 Tuesday, October 4, 2016 Textbook: Sections 7.3, 7.4, 8.1, 8.2, 8.3 Objectives: Identify, and resist the temptation to fall for, the gambler s fallacy Define random variable
More information3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability
Random variable The outcome of each procedure is determined by chance. Probability Distributions Normal Probability Distribution N Chapter 6 Discrete Random variables takes on a countable number of values
More informationEE/CpE 345. Modeling and Simulation. Fall Class 5 September 30, 2002
EE/CpE 345 Modeling and Simulation Class 5 September 30, 2002 Statistical Models in Simulation Real World phenomena of interest Sample phenomena select distribution Probabilistic, not deterministic Model
More informationDiscrete Distributions
Discrete Distributions 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
More informationChapter 2: Probability Part 1
Engineering Probability & Statistics (AGE 1150) Chapter 2: Probability Part 1 Dr. O. Phillips Agboola Sample Space (S) Experiment: is some procedure (or process) that we do and it results in an outcome.
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 informationDiscrete Random Variable
Discrete Random Variable Outcome of a random experiment need not to be a number. We are generally interested in some measurement or numerical attribute of the outcome, rather than the outcome itself. n
More informationChapter 6 Continuous Probability Distributions
Math 3 Chapter 6 Continuous Probability Distributions The observations generated by different statistical experiments have the same general type of behavior. The followings are the probability distributions
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 informationRandom variables (discrete)
Random variables (discrete) Saad Mneimneh 1 Introducing random variables A random variable is a mapping from the sample space to the real line. We usually denote the random variable by X, and a value that
More informationLecture notes for probability. Math 124
Lecture notes for probability Math 124 What is probability? Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments whose result
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 informationMath 1313 Experiments, Events and Sample Spaces
Math 1313 Experiments, Events and Sample Spaces At the end of this recording, you should be able to define and use the basic terminology used in defining experiments. Terminology The next main topic in
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 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 informationMAT X (Spring 2012) Random Variables - Part I
MAT 2379 3X (Spring 2012) Random Variables - Part I While writing my book [Stochastic Processes] I had an argument with Feller. He asserted that everyone said random variable and I asserted that everyone
More informationIntroduction to Probability, Fall 2013
Introduction to Probability, Fall 2013 Math 30530 Section 01 Homework 4 Solutions 1. Chapter 2, Problem 1 2. Chapter 2, Problem 2 3. Chapter 2, Problem 3 4. Chapter 2, Problem 5 5. Chapter 2, Problem 6
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 informationLecture 13. Poisson Distribution. Text: A Course in Probability by Weiss 5.5. STAT 225 Introduction to Probability Models February 16, 2014
Lecture 13 Text: A Course in Probability by Weiss 5.5 STAT 225 Introduction to Probability Models February 16, 2014 Whitney Huang Purdue University 13.1 Agenda 1 2 3 13.2 Review So far, we have seen discrete
More informationRecap. The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY INFERENTIAL STATISTICS
Recap. Probability (section 1.1) The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY Population Sample INFERENTIAL STATISTICS Today. Formulation
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 informationIntroduction to Probability Theory for Graduate Economics Fall 2008
Introduction to Probability Theory for Graduate Economics Fall 008 Yiğit Sağlam October 10, 008 CHAPTER - RANDOM VARIABLES AND EXPECTATION 1 1 Random Variables A random variable (RV) is a real-valued function
More informationSTAT 345 Spring 2018 Homework 4 - Discrete Probability Distributions
STAT 345 Spring 2018 Homework 4 - Discrete Probability Distributions Name: Please adhere to the homework rules as given in the Syllabus. 1. Coin Flipping. Timothy and Jimothy are playing a betting game.
More informationLecture 6 Random Variable. Compose of procedure & observation. From observation, we get outcomes
ENM 07 Lecture 6 Random Variable Random Variable Eperiment (hysical Model) Compose of procedure & observation From observation we get outcomes From all outcomes we get a (mathematical) probability model
More informationCommon Discrete Distributions
Common Discrete Distributions Statistics 104 Autumn 2004 Taken from Statistics 110 Lecture Notes Copyright c 2004 by Mark E. Irwin Common Discrete Distributions There are a wide range of popular discrete
More informationModule 8 Probability
Module 8 Probability Probability is an important part of modern mathematics and modern life, since so many things involve randomness. The ClassWiz is helpful for calculating probabilities, especially those
More informationProbability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin,
Chapter 8 Exercises Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin, morin@physics.harvard.edu 8.1 Chapter 1 Section 1.2: Permutations 1. Assigning seats *
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 informationb. ( ) ( ) ( ) ( ) ( ) 5. Independence: Two events (A & B) are independent if one of the conditions listed below is satisfied; ( ) ( ) ( )
1. Set a. b. 2. Definitions a. Random Experiment: An experiment that can result in different outcomes, even though it is performed under the same conditions and in the same manner. b. Sample Space: This
More informationPage Max. Possible Points Total 100
Math 3215 Exam 2 Summer 2014 Instructor: Sal Barone Name: GT username: 1. No books or notes are allowed. 2. You may use ONLY NON-GRAPHING and NON-PROGRAMABLE scientific calculators. All other electronic
More informationTOPIC 12: RANDOM VARIABLES AND THEIR DISTRIBUTIONS
TOPIC : RANDOM VARIABLES AND THEIR DISTRIBUTIONS In the last section we compared the length of the longest run in the data for various players to our expectations for the longest run in data generated
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 informationGuidelines for Solving Probability Problems
Guidelines for Solving Probability Problems CS 1538: Introduction to Simulation 1 Steps for Problem Solving Suggested steps for approaching a problem: 1. Identify the distribution What distribution does
More information9/6/2016. Section 5.1 Probability. Equally Likely Model. The Division Rule: P(A)=#(A)/#(S) Some Popular Randomizers.
Chapter 5: Probability and Discrete Probability Distribution Learn. Probability Binomial Distribution Poisson Distribution Some Popular Randomizers Rolling dice Spinning a wheel Flipping a coin Drawing
More informationProbability. Chapter 1 Probability. A Simple Example. Sample Space and Probability. Sample Space and Event. Sample Space (Two Dice) Probability
Probability Chapter 1 Probability 1.1 asic Concepts researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people
More informationCHAPTER 3 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. 3.1 Concept of a Random Variable. 3.2 Discrete Probability Distributions
CHAPTER 3 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 3.1 Concept of a Random Variable Random Variable A random variable is a function that associates a real number with each element in the sample space.
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 informationMarquette University Executive MBA Program Statistics Review Class Notes Summer 2018
Marquette University Executive MBA Program Statistics Review Class Notes Summer 2018 Chapter One: Data and Statistics Statistics A collection of procedures and principles
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 informationTopic 3: Introduction to Probability
Topic 3: Introduction to Probability 1 Contents 1. Introduction 2. Simple Definitions 3. Types of Probability 4. Theorems of Probability 5. Probabilities under conditions of statistically independent events
More informationLecture 2: Probability and Distributions
Lecture 2: Probability and Distributions Ani Manichaikul amanicha@jhsph.edu 17 April 2007 1 / 65 Probability: Why do we care? Probability helps us by: Allowing us to translate scientific questions info
More informationGreat Theoretical Ideas in Computer Science
15-251 Great Theoretical Ideas in Computer Science Probability Theory: Counting in Terms of Proportions Lecture 10 (September 27, 2007) Some Puzzles Teams A and B are equally good In any one game, each
More information