Binomial distribution
|
|
- Theodora Willis
- 5 years ago
- Views:
Transcription
1 Binomial distribution Jon Michael Gran Department of Biostatistics, UiO MF9130 Introductory course in statistics Tuesday / 28
2 Overview Binomial distribution (Aalen chapter 4, Kirkwood and Sterne chapter ) Probability distributions for counting variables Binomial distribution Introduction to hypothesis testing 2 / 28
3 Probability distributions for counting variables Concepts Random (stochastic) trial: In forehand, we don t know the outcome, but we know the set of possible outcomes Random variable: Number linked to the outcome. We don t know this value before the trial is carried out Probability distribution: The set of probabilities for each of the possible values 3 / 28
4 Sucesses and Failures Often, a process has only two outcomes. A few examples can be: When tossing a coin, you get either head or tails An industrial process produces a product that can be either usable or defect A HIV test look for the presence or absence of antibodies in the blood 4 / 28
5 Sucesses and Failures Often, a process has only two outcomes. A few examples can be: When tossing a coin, you get either head or tails An industrial process produces a product that can be either usable or defect A HIV test look for the presence or absence of antibodies in the blood Or, there are just two outcomes of interest: Throwing a dice, you are only interested in if you get a six or not 4 / 28
6 Binomial trials A series of random trials satisfying the following requirements: In each trial one registers whether an event A occurs or not The probability of A is the same in each trial, and is denoted by p All trials are independent of each other 5 / 28
7 Examples: Binomial trials Tossing a coin: A: Tails, p = 1/2 Throwing a dice: A: Six, p = 1/6 Child births: A: Girl, p = 0.5 A: Spina bifida (ryggmargsbrokk), p = A: Birth weight < 2500, p =? Epidemiology: A: Myocardial infarction, p =? Genetics: Mother and father are carriers of the gene for cystic fibrosis: A: Child ill, p = 1/4 6 / 28
8 Concepts Counting variables: How many times does event A occur in a series of trials? Discrete variables (measured only by means of whole numbers) Probability distributions for counting variables: What is the probability for each possible number of events A? 7 / 28
9 The probability for a certain sequence Say you do n trials, looking for an event A to occur with probability p in each trial The result is a sequence like A, Ā, Ā, A, Ā, A, A, Ā,..., A Now say that A take place x times, meaning n x occurrences of Ā What is the probability of such a sequence? 8 / 28
10 The probability for a certain sequence Say you do n trials, looking for an event A to occur with probability p in each trial The result is a sequence like A, Ā, Ā, A, Ā, A, A, Ā,..., A Now say that A take place x times, meaning n x occurrences of Ā What is the probability of such a sequence? Recall that probabilities for independent events may be multiplied! P(sequence above) = p(1 p)(1 p)p(1 p)pp(1 p)...p x number of p and n x number of (1 p): P(given sequence) = p x (1 p) n x 8 / 28
11 The probability for a certain sequence Say you do n trials, looking for an event A to occur with probability p in each trial The result is a sequence like A, Ā, Ā, A, Ā, A, A, Ā,..., A Now say that A take place x times, meaning n x occurrences of Ā What is the probability of such a sequence? Recall that probabilities for independent events may be multiplied! P(sequence above) = p(1 p)(1 p)p(1 p)pp(1 p)...p x number of p and n x number of (1 p): P(given sequence) = p x (1 p) n x But, you can get x successes out of n trials in many different orders. What about the probability of x? 8 / 28
12 The number of non-ordered selections Want to find the number of ways that x objects can be chosen from a total of n objects, regardless of order 9 / 28
13 The number of non-ordered selections Want to find the number of ways that x objects can be chosen from a total of n objects, regardless of order This number is given by the binomial coefficient ( n) x 9 / 28
14 The number of non-ordered selections Want to find the number of ways that x objects can be chosen from a total of n objects, regardless of order This number is given by the binomial coefficient ( n) x ( n x) = n (n 1) (n 2)... (n x+1) x! = n! x!(n x)!, where n (n 1) (n 2)... (n x + 1) is the number of ordered selections when picking x objects out of a total of n objects x! = x (x 1) (x 2) is the number of ways to order s objects (or the number of permutations of s objects) For example: ( 4) 3 = = 4, and ( 10) 4 = = / 28
15 Three ways to calculate the binomial coefficient ( ) n x Use the formula ( n) x = n (n 1) (n 2)... (n x+1) x! = n! x!(n x)! where n! = (n 1) n, and 0! = 1 10 / 28
16 Three ways to calculate the binomial coefficient ( ) n x Use the formula ( n) x = n (n 1) (n 2)... (n x+1) x! = n! x!(n x)! where n! = (n 1) n, and 0! = 1 Use a calculator or computer. Usually the notation is ( n ) x = ncx(n, x) 10 / 28
17 Three ways to calculate the binomial coefficient ( ) n x Use the formula ( n) x = n (n 1) (n 2)... (n x+1) x! = n! x!(n x)! where n! = (n 1) n, and 0! = 1 Use a calculator or computer. Usually the notation is ( n ) x = ncx(n, x) Use Pascal s triangle (see next slide) 10 / 28
18 Pascal s triangle ( n x) refers to row n, place number x + 1 from the left 11 / 28
19 Pascal s triangle ( n x) refers to row n, place number x + 1 from the left It can be shown directly from the definition (formula) of the binomial coefficient that ( n) ( x + n ) ( x+1 = n+1 ) x+1 11 / 28
20 Summary of the last four slides... We looked for an event A with probability p in n trials and got a sequence A, Ā, Ā, A, Ā, A, A, Ā,..., A. In this sequence, A took place x times and Ā took place (n x) times. The probability of that particular sequence was p x (1 p) n x The sequence can be orded in many different ways, all with the same probability The number of ways to sort the sequence is given by ( n) x We can now derive (or have already) the binomial distribution function, that is, the probability P(x) for every possible x. 12 / 28
21 Binomial distribution Binomial probability distribution We observe n trials. The probability that A occurs exactly x times is given by P(X = x) = ( n x) p x (1 p) n x, or P(X = x) = the number of ways to distribute x events A in a sequence of length n the probability of one particular sequence with x events A 13 / 28
22 Example: The binomial distribution for n = 8 and p = 0.15 Often written Binomial(n=8, p=0.15), or Bin(8,0.15) 14 / 28
23 Properties of probability distributions in general If you sum or integrate over all possible outcomes for a probability distribution, you should get 1 Not surprising, from the probability theory lecture! Most probability distributions have an expected value (corresponding to the mean) and a variance (or standard deviation) For the binomial distribution: E(X) = np Var(X) = np(1 p) 15 / 28
24 Example: binomial distribution for n = 8 and p = 0.15 (cont.) Let s say p = 0.15 is the probability that a person signing up to test for a certain disease get a positive result. A certain day you are going to do n = 8 such tests. What is the expected number of positive tests (the mean)? What is the variance and the standard deviation? What is the probability that you get 2 or more positives? 16 / 28
25 Histograms from a binomial distribution with n = 8 trials, and four different values of p 17 / 28
26 Example: Blood type (without using the formula) Consider three randomly sampled individuals. How many have blood type A? Independent individuals (random sampling) The only outcomes are bloodtype A or not bloodtype A Constant probabilities: P(A) = 0.4 and P(not A) = = / 28
27 All possible combinations for the three persons: Person 1 Person 2 Person 3 Probability A A A = A A Not A = A Not A A = Not A A A = A Not A Not A = Not A A Not A = Not A Not A A = Not A Not A Not A = = 1 19 / 28
28 The binomial distribution for the number of people with blood type A is then: Number of people Probability with blood type A = = = 1 20 / 28
29 Example: Family with four kids We re looking at a family with four kids, with no monozygotic twins, so the gender of the kids are (approx.) independent. The probability of getting a boy in Norway is What is the probability that two of the kids are boys? P(X = 2) = ( 4) = = / 28
30 When the gender distribution of 7745 American families are given, together with the probability distribution from the binomial distribution we get this table: We observe a good agreement! 22 / 28
31 Example: Multiple choice exam Suppose a person knows absolutely nothing, and simply guesses on a 12 item test with three alternatives. What is their probability of passing the test, if 65% is the passing mark? You need 8 right to pass: = 7.8 P(X = 8) = ( 12 8 ) = 0.15 P(X = 9) = ( 12 9 ) = P(X = 10) = ( 12 10) = P(X = 11) = ( 12 11) = P(X = 12) = ( 12) = The probability to pass is P(X 8) / 28
32 Introduction to hypothesis testing Statistical hypothesis testing A method to draw conclusions from uncertain data, estimating the uncertainty in the conclusion Based on the computation of a particular probability, called the P value (The probability for the given result or a more extreme result to occur) We introduce hypothesis testing trough an example related to the binomial distribution (the binomial test). More about Hypothesis testing in general tomorrow. 24 / 28
33 Example: Clinical trial Want to try out a new medicine against migraine. Let s denote the new medicine N and the traditional one T. Each patient get N one month, and T another month. We want to find out which month the patient feels the best/have less migraine. The trial is randomized and made blind to make it fair. Cross over study: Eight patients try both medications in randomized order 25 / 28
34 Say that 7 of totally 8 patients preferred N Is N better than T? Let p be the probability of a patient preferring N Null hypothesis: H 0 : p = 1/2 (Both treatments equally good) Alternative hypothesis: H A : p > 1/2 (N is better) 26 / 28
35 If the null hypothesis holds, X is binomially distributed with n = 8 and p = 1/2 P value: P(X 7 H 0 ) = ( 8) 7 (1/2)7 (1/2) 1 + ( 8) 8 (1/2)8 (1/2) 0 = = 3.5% Mindset: If P is small, this indicates that H 0 is not likely to be true We reject H 0 and accept H A if P is smaller than the level of significance. This is often set to 5%, or sometimes 1% or less With a 5% level of significance we would reject H 0 and say that N is a better drug than T 27 / 28
36 Summary Key words (Discrete) Probability distributions Binomial trials Counting variables The binomial coefficient The binomial distribution Notation x! ( n) x P(X = x) 28 / 28
Chapter 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 informationProblem A Grade x P(x) To get "C" 1 or 2 must be 1 0.05469 B A 2 0.16410 3 0.27340 4 0.27340 5 0.16410 6 0.05470 7 0.00780 0.2188 0.5468 0.2266 Problem B Grade x P(x) To get "C" 1 or 2 must 1 0.31150 be
More informationguessing Bluman, Chapter 5 2
Bluman, Chapter 5 1 guessing Suppose there is multiple choice quiz on a subject you don t know anything about. 15 th Century Russian Literature; Nuclear physics etc. You have to guess on every question.
More informationProbability & Statistics Chapter 5: Binomial Distribution
Probability & Statistics Chapter 5: Binomial Distribution Notes and Examples Binomial Distribution When a variable can be viewed as having only two outcomes, call them success and failure, it may be considered
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 informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic Probability Distributions: Binomial and Poisson Distributions Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationLecture 6 Probability
Faculty of Medicine Epidemiology and Biostatistics الوبائيات واإلحصاء الحيوي (31505204) Lecture 6 Probability By Hatim Jaber MD MPH JBCM PhD 3+4-7-2018 1 Presentation outline 3+4-7-2018 Time Introduction-
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 informationThe Binomial Distribution
The Binomial Distribution Patrick Breheny February 16 Patrick Breheny STA 580: Biostatistics I 1/38 Random variables The Binomial Distribution Random variables The binomial coefficients The binomial distribution
More informationchapter 13: Binomial Distribution Exercises (binomial)13.6, 13.12, 13.22, 13.43
chapter 13: Binomial Distribution ch13-links binom-tossing-4-coins binom-coin-example ch13 image Exercises (binomial)13.6, 13.12, 13.22, 13.43 CHAPTER 13: Binomial Distributions The Basic Practice of Statistics
More information4.2 Bernoulli Trials and Binomial Distributions
Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan 4.2 Bernoulli Trials and Binomial Distributions A Bernoulli trial 1 is an experiment with exactly two outcomes: Success and
More informationSTAT 111 Recitation 2
STAT 111 Recitation 2 Linjun Zhang October 10, 2017 Misc. Please collect homework 1 (graded). 1 Misc. Please collect homework 1 (graded). Office hours: 4:30-5:30pm every Monday, JMHH F86. 1 Misc. Please
More informationChapter 8: Binomial and Geometric Distributions
Chapter 8: Binomial and Geometric Distributions Section 8.1 Binomial Distributions The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Section 8.1 Binomial Distribution Learning Objectives
More informationThe Binomial distribution
The Binomial distribution Examples and Definition Binomial Model (an experiment ) 1 A series of n independent trials is conducted. 2 Each trial results in a binary outcome (one is labeled success the other
More informationSampling Distributions For Counts and Proportions
Sampling Distributions For Counts and Proportions IPS Chapter 5.1 2009 W. H. Freeman and Company Objectives (IPS Chapter 5.1) Sampling distributions for counts and proportions Binomial distributions for
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 informationEXERCISES ACTIVITY 6.7
762 CHAPTER 6 PROBABILITY MODELS EXERCISES ACTIVITY 6.7 1. Compute each of the following: 100! a. 5! I). 98! c. 9P 9 ~~ d. np 9 g- 8Q e. 10^4 6^4 " 285^1 f-, 2 c 5 ' sq ' sq 2. How many different ways
More informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.3 Binomial and Geometric Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Binomial and Geometric Random
More informationChapter 6: Random Variables
Chapter 6: Random Variables Section 6.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6.2 Transforming and
More informationPart 10: The Binomial Distribution
Part 10: The Binomial Distribution The binomial distribution is an important example of a probability distribution for a discrete random variable. It has wide ranging applications. One readily available
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 informationMath 14 Lecture Notes Ch. 4.3
4.3 The Binomial Distribution Example 1: The former Sacramento King's DeMarcus Cousins makes 77% of his free throws. If he shoots 3 times, what is the probability that he will make exactly 0, 1, 2, or
More information***SECTION 8.1*** The Binomial Distributions
***SECTION 8.1*** The Binomial Distributions CHAPTER 8 ~ The Binomial and Geometric Distributions In practice, we frequently encounter random phenomenon where there are two outcomes of interest. For example,
More informationMath 361. Day 8 Binomial Random Variables pages 27 and 28 Inv Do you have ESP? Inv. 1.3 Tim or Bob?
Math 361 Day 8 Binomial Random Variables pages 27 and 28 Inv. 1.2 - Do you have ESP? Inv. 1.3 Tim or Bob? Inv. 1.1: Friend or Foe Review Is a particular study result consistent with the null model? Learning
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 informationRandom Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES
Random Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES Essential Question How can I determine whether the conditions for using binomial random variables are met? Binomial Settings When the
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 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 informationSTOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions 5/31/11 Lecture 14 1 Statistic & Its Sampling Distribution
More informationProbability Distributions
Chapter 6 Discrete Probability Distributions Section 6-2 Probability Distributions Definitions Let S be the sample space of a probability experiment. A random variable X is a function from the set S into
More informationE509A: Principle of Biostatistics. GY Zou
E509A: Principle of Biostatistics (Week 2: Probability and Distributions) GY Zou gzou@robarts.ca Reporting of continuous data If approximately symmetric, use mean (SD), e.g., Antibody titers ranged from
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 6 Random Variables
CHAPTER 6 Random Variables 6.3 Binomial and Geometric Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Binomial and Geometric Random
More informationExperimental Probability - probability measured by performing an experiment for a number of n trials and recording the number of outcomes
MDM 4U Probability Review Properties of Probability Experimental Probability - probability measured by performing an experiment for a number of n trials and recording the number of outcomes Theoretical
More informationChapter 15 - The Binomial Formula PART
Chapter 15 - The Binomial Formula PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 15 - The Binomial Formula 1 / 19 Pascal s Triangle In this chapter we explore
More informationSection 6.3 Binomial and Geometric Random Variables
Section 6.3 Binomial and Geometric Random Variables Mrs. Daniel AP Stats Binomial Settings A binomial setting arises when we perform several independent trials of the same chance process and record 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 informationOverview. Definitions. Definitions. Graphs. Chapter 4 Probability Distributions. probability distributions
Chapter 4 Probability Distributions 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 The Poisson Distribution
More informationThe binomial distribution p314
The binomial distribution p314 Example: A biased coin (P(H) = p = 0.6) ) is tossed 5 times. Let X be the number of H s. Fine P(X = 2). This X is a binomial r. v. The binomial setting p314 1. There are
More informationThe Binomial Distribution
Patrick Breheny February 21 Patrick Breheny University of Iowa Introduction to Biostatistics (BIOS 4120) 1 / 16 So far, we have discussed the probability of single events In research, however, the data
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 informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.3 Binomial and Geometric Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers 6.3 Reading Quiz (T or F) 1.
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 informationRandom variables The binomial distribution The normal distribution Other distributions. Distributions. Patrick Breheny.
Distributions February 11 Random variables Anything that can be measured or categorized is called a variable If the value that a variable takes on is subject to variability, then it the variable is a random
More informationExample 1: Identify the following random variables as discrete or continuous: a) Weight of a package. b) Number of students in a first-grade classroom
Section 5-1 Probability Distributions I. Random Variables A variable x is a if the value that it assumes, corresponding to the of an experiment, is a or event. A random variable is if it potentially can
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 informationMANAGEMENT PRINCIPLES AND STATISTICS (252 BE)
MANAGEMENT PRINCIPLES AND STATISTICS (252 BE) Normal and Binomial Distribution Applied to Construction Management Sampling and Confidence Intervals Sr Tan Liat Choon Email: tanliatchoon@gmail.com Mobile:
More informationMATH1215: Mathematical Thinking Sec. 08 Spring Worksheet 9: Solution. x P(x)
N. Name: MATH: Mathematical Thinking Sec. 08 Spring 0 Worksheet 9: Solution Problem Compute the expected value of this probability distribution: x 3 8 0 3 P(x) 0. 0.0 0.3 0. Clearly, a value is missing
More informationChapter 11. Data Descriptions and Probability Distributions. Section 4 Bernoulli Trials and Binomial Distribution
Chapter 11 Data Descriptions and Probability Distributions Section 4 Bernoulli Trials and Binomial Distribution 1 Learning Objectives for Section 11.4 Bernoulli Trials and Binomial Distributions The student
More informationProbability Distribution Unit Review
Probability Distribution Unit Review Topics: Pascal's Triangle and Binomial Theorem Probability Distributions and Histograms Expected Values, Fair Games of chance Binomial Distributions Hypergeometric
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 informationSTT315 Chapter 4 Random Variables & Probability Distributions AM KM
Before starting new chapter: brief Review from Algebra Combinations In how many ways can we select x objects out of n objects? In how many ways you can select 5 numbers out of 45 numbers ballot to win
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 2 Discrete Distributions The binomial distribution 1 Chapter 2 Discrete Distributions Bernoulli trials and the
More informationMath 160 Professor Busken Chapter 5 Worksheets
Math 160 Professor Busken Chapter 5 Worksheets Name: 1. Find the expected value. Suppose you play a Pick 4 Lotto where you pay 50 to select a sequence of four digits, such as 2118. If you select the same
More informationBinomial Random Variable - The count X of successes in a binomial setting
6.3.1 Binomial Settings and Binomial Random Variables What do the following scenarios have in common? Toss a coin 5 times. Count the number of heads. Spin a roulette wheel 8 times. Record how many times
More information5.1 Personal Probability
5. Probability Value Page 1 5.1 Personal Probability Although we think probability is something that is confined to math class, in the form of personal probability it is something we use to make decisions
More informationThe Binomial and Geometric Distributions. Chapter 8
The Binomial and Geometric Distributions Chapter 8 8.1 The Binomial Distribution A binomial experiment is statistical experiment that has the following properties: The experiment consists of n repeated
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 informationThe Binomial Distribution
AQR Reading: Binomial Probability Reading #1: The Binomial Distribution A. It would be very tedious if, every time we had a slightly different problem, we had to determine the probability distributions
More informationRandom variables The binomial distribution The normal distribution Sampling distributions. Distributions. Patrick Breheny.
Distributions September 17 Random variables Anything that can be measured or categorized is called a variable If the value that a variable takes on is subject to variability, then it the variable is a
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 information8.1 Binomial Distributions
8.1 Binomial Distributions The Binomial Setting The 4 Conditions of a Binomial Setting: 1.Each observation falls into 1 of 2 categories ( success or fail ) 2 2.There is a fixed # n of observations. 3.All
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 informationSTOR Lecture 7. Random Variables - I
STOR 435.001 Lecture 7 Random Variables - I Shankar Bhamidi UNC Chapel Hill 1 / 31 Example 1a: Suppose that our experiment consists of tossing 3 fair coins. Let Y denote the number of heads that appear.
More informationASSIGNMENT 14 section 10 in the probability and statistics module
McMaster University Math 1LT3 ASSIGNMENT 14 section 10 in the probability and statistics module 1. (a) A shipment of 2,000 containers has arrived at the port of Vancouver. As part of the customs inspection,
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 8: The Binomial and Geometric Distributions
Chapter 8: The Binomial and Geometric Distributions 8.1 Binomial Distributions 8.2 Geometric Distributions 1 Let me begin with an example My best friends from Kent School had three daughters. What is the
More information6.1 Binomial Theorem
Unit 6 Probability AFM Valentine 6.1 Binomial Theorem Objective: I will be able to read and evaluate binomial coefficients. I will be able to expand binomials using binomial theorem. Vocabulary Binomial
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 informationFixed number of n trials Independence
The Binomial Setting Binomial Distributions IB Math SL - Santowski Fixed number of n trials Independence Two possible outcomes: success or failure Same probability of a success for each observation If
More information6. THE BINOMIAL DISTRIBUTION
6. THE BINOMIAL DISTRIBUTION Eg: For 1000 borrowers in the lowest risk category (FICO score between 800 and 850), what is the probability that at least 250 of them will default on their loan (thereby rendering
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationLesson 97 - Binomial Distributions IBHL2 - SANTOWSKI
Lesson 97 - Binomial Distributions IBHL2 - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin 3 times where P(H) = / (b) THUS, find the probability
More informationOpening Exercise: Lesson 91 - Binomial Distributions IBHL2 - SANTOWSKI
08-0- Lesson 9 - Binomial Distributions IBHL - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin times where P(H) = / (b) THUS, find the probability
More informationLecture 3. Sampling distributions. Counts, Proportions, and sample mean.
Lecture 3 Sampling distributions. Counts, Proportions, and sample mean. Statistical Inference: Uses data and summary statistics (mean, variances, proportions, slopes) to draw conclusions about a population
More informationBernoulli and Binomial Distributions
Bernoulli and Binomial Distributions Bernoulli Distribution a flipped coin turns up either heads or tails an item on an assembly line is either defective or not defective a piece of fruit is either damaged
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 informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
More informationTheoretical Foundations
Theoretical Foundations Probabilities Monia Ranalli monia.ranalli@uniroma2.it Ranalli M. Theoretical Foundations - Probabilities 1 / 27 Objectives understand the probability basics quantify random phenomena
More informationS = 1,2,3, 4,5,6 occurs
Chapter 5 Discrete Probability Distributions The observations generated by different statistical experiments have the same general type of behavior. Discrete random variables associated with these experiments
More informationChapter 4 Discrete Random variables
Chapter 4 Discrete Random variables A is a variable that assumes numerical values associated with the random outcomes of an experiment, where only one numerical value is assigned to each sample point.
More informationProbability and Statistics
Probability and Statistics Alvin Lin Probability and Statistics: January 2017 - May 2017 Binomial Random Variables There are two balls marked S and F in a basket. Select a ball 3 times with replacement.
More informationThe Bernoulli distribution
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationChapter 5 Probability Distributions. Section 5-2 Random Variables. Random Variable Probability Distribution. Discrete and Continuous Random Variables
Chapter 5 Probability Distributions Section 5-2 Random Variables 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance and Standard Deviation for the Binomial Distribution Random
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationPROBABILITY AND STATISTICS, A16, TEST 1
PROBABILITY AND STATISTICS, A16, TEST 1 Name: Student number (1) (1.5 marks) i) Let A and B be mutually exclusive events with p(a) = 0.7 and p(b) = 0.2. Determine p(a B ) and also p(a B). ii) Let C and
More informationLecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances
Physical Principles in Biology Biology 3550 Fall 2018 Lecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances Monday, 10 September 2018 c David P. Goldenberg University
More informationMA 1125 Lecture 12 - Mean and Standard Deviation for the Binomial Distribution. Objectives: Mean and standard deviation for the binomial distribution.
MA 5 Lecture - Mean and Standard Deviation for the Binomial Distribution Friday, September 9, 07 Objectives: Mean and standard deviation for the binomial distribution.. Mean and Standard Deviation of the
More informationEx 1) Suppose a license plate can have any three letters followed by any four digits.
AFM Notes, Unit 1 Probability Name 1-1 FPC and Permutations Date Period ------------------------------------------------------------------------------------------------------- The Fundamental Principle
More information2. Modeling Uncertainty
2. Modeling Uncertainty Models for Uncertainty (Random Variables): Big Picture We now move from viewing the data to thinking about models that describe the data. Since the real world is uncertain, our
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 informationChapter 4 Discrete Random variables
Chapter 4 Discrete Random variables A is a variable that assumes numerical values associated with the random outcomes of an experiment, where only one numerical value is assigned to each sample point.
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability
More informationMidterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
More informationChapter 3 - Lecture 3 Expected Values of Discrete Random Va
Chapter 3 - Lecture 3 Expected Values of Discrete Random Variables October 5th, 2009 Properties of expected value Standard deviation Shortcut formula Properties of the variance Properties of expected value
More informationChapter 8 Additional Probability Topics
Chapter 8 Additional Probability Topics 8.6 The Binomial Probability Model Sometimes experiments are simulated using a random number function instead of actually performing the experiment. In Problems
More informationMath Week in Review #10. Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
Math 141 Spring 2006 c Heather Ramsey Page 1 Section 8.4 - Binomial Distribution Math 141 - Week in Review #10 Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
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 informationSTAT Mathematical Statistics
STAT 6201 - Mathematical Statistics Chapter 3 : Random variables 5, Event, Prc ) Random variables and distributions Let S be the sample space associated with a probability experiment Assume that we have
More information