HUDM4122 Probability and Statistical Inference. February 23, 2015
|
|
- Samantha Carson
- 5 years ago
- Views:
Transcription
1 HUDM4122 Probability and Statistical Inference February 23, 2015
2 In the last class We studied Bayes Theorem and the Law of Total Probability
3 Any questions or comments?
4 Today Chapter 4.8 in Mendenhall, Beaver, & Beaver Probability Distributions Random Variables
5 Random Variable A variable x is a random variable if the value that it assumes, corresponding to the outcome of an experiment, is a chance or random event. MBB, p.158 A random variable is a variable whose value is subject to variations due to chance. -- Wikipedia
6 Random Variable It is not that the variable can have any value at random But that the variable s value comes from random sampling
7 These are random variables A coin flip s result Number of times a randomly selected student is sent to the principal s office NY Regents Exam score for a randomly selected student in NY State Number of people on the subway at a randomly selected time
8 These are not random variables I flip a biased coin that gives 100% heads The temperature setting on your oven you set it yourself These values are not subject to chance
9 A random variable s value You can never say for sure what it s going to be There is a certainly probability that it will have certain values
10 Questions? Comments?
11 Reminder Discrete variable Can have limited number of values Continuous/numerical variable Can have infinite number of values
12 Which of these are Discrete Variables? Number of heads in 3 coin flips Sum of rolling two 6-sided dice Temperature outside How late is 2 train Height of a person, rounded to closest inch Number of times a randomly selected student is sent to the principal s office
13 Probability Distribution A probability distribution for random variable X gives the possible values of X, x 1 x n And the probability p(x i ) associated with each value of X
14 Probability Distribution A probability distribution for random variable X gives the possible values of X, x 1 x n And the probability p(x i ) associated with each value of X Each value of X is mutually exclusive The sum of p(x i ) adds to 1
15 Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2
16 Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x P(x)
17 Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x P(x) 0 1/4 1 2/4 2 1/4
18 Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x P(x) 0 1/4 1 1/2 2 1/4
19 Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x P(x)
20 Probability Histogram x P(x) p(x) x
21 You try it I flip a coin three times What is the probability distribution on the number of heads?
22 You try it: What is the Probability Distribution? I collected the following data on the temperature in NYC Day Temp Day Temp 1 Cold 9 Cold 2 Cold 10 Cold 3 Freezing 11 Cold 4 Cold 12 Not That Bad 5 Cold 13 Cold 6 Freezing 14 F ing Freezing 7 F ing Freezing 15 F ing Freezing 8 Freezing 16 Cold
23 Note that probability distributions can have many values p(x) x
24 Later in the semester We ll talk about probability distributions for continuous variables
25 Questions? Comments?
26 Expected Value of Probability Distribution The value you can expect to get on average if you re-run your experiment many times Take each value, multiply it by its probability Add those together E(x) = ( )
27 Expected Value of Probability Distribution The value you can expect to get on average if you re-run your experiment many times Take each value, multiply it by its probability Add those together E(x) = ( ) This is also called the mean of the random variable, µ
28 Example x P(x) (0.25) + 1(0.5) + 2(0.25)
29 Example x P(x) = 1
30 You can expect 1 head on average if you flip a coin twice, infinite times x P(x) = 1
31 You Try It x P(x)
32 You Try It x P(x) (0.1)+1(0.5)+2(0.2)+3(0.2)
33 You Try It x P(x) =
34 You Try It My dad s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket One time he won $1000 What is the expected value for playing the lottery?
35 You Try It My dad s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket One time he won $1000 What is the expected value for playing the lottery? (1/14600)(1000) + (14599/14600)(-1)
36 You Try It My dad s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket One time he won $1000 What is the expected value for playing the lottery? (1/14600)(1000) + (14599/14600)(-1)= = $-0.931
37 Questions? Comments?
38 Variance of Discrete Random Variable ( )
39 Standard Deviation of Discrete Random Variable ( )
40 Example: SD of Random Variable x P(x)
41 Example: SD of Random Variable x P(x) Recall: µ =
42 Example: xsd of Random P(x) Variable Recall: µ = 1 ( )
43 Example: xsd of Random P(x) Variable Recall: µ = (0.25)
44 Example: xsd of Random P(x) Variable Recall: µ = 1 1(0.25)
45 Example: xsd of Random P(x) Variable Recall: µ =
46 Example: xsd of Random P(x) Variable Recall: µ =
47 You Try It: SD of random variable x P(x) µ = 1.5
48 Any last comments or questions for the day?
49 If there s time: Do this in solver-explainer pairs Practice with extended version of Bayes Rule ( ) = ( ) )
50 Example There are three professors who teach HUDM4122. Let s call them A, B, and C P(student is in prof A s class) = P(A) = 0.4 P(student is in prof B s class) = P(B) = 0.3 P(student is in prof C s class) = P(C) = 0.1
51 Example P(student is in prof A s class) = P(A) = 0.4 P(student is in prof B s class) = P(B) = 0.3 P(student is in prof C s class) = P(C) = 0.1 P(learned stats A)=0.8 P(learned stats B)=0.6 P(learned stats C)=0.4
52 What is P(A learned stats)? P(student is in prof A s class) = P(A) = 0.4 P(student is in prof B s class) = P(B) = 0.3 P(student is in prof C s class) = P(C) = 0.1 P(learned stats A)=0.8 P(learned stats B)=0.6 P(learned stats C)=0.4
53 Upcoming Classes 2/25 Binomial Probability Distribution Ch. 5-2 HW 4 due 2/28 Normal Probability Distribution
54 Homework 4 Due in 2 days In the ASSISTments system
HUDM4122 Probability and Statistical Inference. March 4, 2015
HUDM4122 Probability and Statistical Inference March 4, 2015 First things first The Exam Due to Monday s class cancellation Today s lecture on the Normal Distribution will not be covered on the Midterm
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 3: Probability Distributions and Statistics
Chapter 3: Probability Distributions and Statistics Section 3.-3.3 3. Random Variables and Histograms A is a rule that assigns precisely one real number to each outcome of an experiment. We usually denote
More informationProbability: Week 4. Kwonsang Lee. University of Pennsylvania February 13, 2015
Probability: Week 4 Kwonsang Lee University of Pennsylvania kwonlee@wharton.upenn.edu February 13, 2015 Kwonsang Lee STAT111 February 13, 2015 1 / 21 Probability Sample space S: the set of all possible
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
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 informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationMATH 118 Class Notes For Chapter 5 By: Maan Omran
MATH 118 Class Notes For Chapter 5 By: Maan Omran Section 5.1 Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Ex1: The test scores
More informationStatistical Methods for NLP LT 2202
LT 2202 Lecture 3 Random variables January 26, 2012 Recap of lecture 2 Basic laws of probability: 0 P(A) 1 for every event A. P(Ω) = 1 P(A B) = P(A) + P(B) if A and B disjoint Conditional probability:
More informationMean, Variance, and Expectation. Mean
3 Mean, Variance, and Expectation The mean, variance, and standard deviation for a probability distribution are computed differently from the mean, variance, and standard deviation for samples. This section
More informationAP Stats ~ Lesson 6B: Transforming and Combining Random variables
AP Stats ~ Lesson 6B: Transforming and Combining Random variables OBJECTIVES: DESCRIBE the effects of transforming a random variable by adding or subtracting a constant and multiplying or dividing by a
More informationX P(X) (c) Express the event performing at least two tests in terms of X and find its probability.
AP Stats ~ QUIZ 6 Name Period 1. The probability distribution below is for the random variable X = number of medical tests performed on a randomly selected outpatient at a certain hospital. X 0 1 2 3 4
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 informationRandom Variables. 6.1 Discrete and Continuous Random Variables. Probability Distribution. Discrete Random Variables. Chapter 6, Section 1
6.1 Discrete and Continuous Random Variables Random Variables A random variable, usually written as X, is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types
More informationProf. Thistleton MAT 505 Introduction to Probability Lecture 3
Sections from Text and MIT Video Lecture: Sections 2.1 through 2.5 http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systemsanalysis-and-applied-probability-fall-2010/video-lectures/lecture-1-probability-models-and-axioms/
More informationSection Random Variables and Histograms
Section 3.1 - Random Variables and Histograms Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationCentral Limit Theorem
Central Limit Theorem Lots of Samples 1 Homework Read Sec 6-5. Discussion Question pg 329 Do Ex 6-5 8-15 2 Objective Use the Central Limit Theorem to solve problems involving sample means 3 Sample Means
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 informationEvery data set has an average and a standard deviation, given by the following formulas,
Discrete Data Sets A data set is any collection of data. For example, the set of test scores on the class s first test would comprise a data set. If we collect a sample from the population we are interested
More informationStatistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
More 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 informationChapter 7: Random Variables
Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables 7.2 Means and Variances of Random Variables 1 Introduction A random variable is a function that associates a unique numerical value
More informationThe Central Limit Theorem. Sec. 8.2: The Random Variable. it s Distribution. it s Distribution
The Central Limit Theorem Sec. 8.1: The Random Variable it s Distribution Sec. 8.2: The Random Variable it s Distribution X p and and How Should You Think of a Random Variable? Imagine a bag with numbers
More informationLECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE
LECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE MSc Đào Việt Hùng Email: hungdv@tlu.edu.vn Random Variable A random variable is a function that assigns a real number to each outcome in the sample space of a
More informationAP Statistics Review Ch. 6
AP Statistics Review Ch. 6 Name 1. Which of the following data sets is not continuous? a. The gallons of gasoline in a car. b. The time it takes to commute in a car. c. Number of goals scored by a hockey
More informationDiscrete Random Variables
Discrete Random Variables MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives During this lesson we will learn to: distinguish between discrete and continuous
More informationDiscrete Random Variables
Discrete Random Variables MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Objectives During this lesson we will learn to: distinguish between discrete and continuous
More informationValue (x) probability Example A-2: Construct a histogram for population Ψ.
Calculus 111, section 08.x The Central Limit Theorem notes by Tim Pilachowski If you haven t done it yet, go to the Math 111 page and download the handout: Central Limit Theorem supplement. Today s lecture
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 informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Ruomeng Lan Texas A&M University October 15, 2014 Ruomeng Lan (TAMU) Math 166 October 15, 2014 1 / 12 Mean, Median and Mode Definition: 1. The average or
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter7 Probability Distributions and Statistics Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number of boys in
More informationStatistics, Their Distributions, and the Central Limit Theorem
Statistics, Their Distributions, and the Central Limit Theorem MATH 3342 Sections 5.3 and 5.4 Sample Means Suppose you sample from a popula0on 10 0mes. You record the following sample means: 10.1 9.5 9.6
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationTOPIC: PROBABILITY DISTRIBUTIONS
TOPIC: PROBABILITY DISTRIBUTIONS There are two types of random variables: A Discrete random variable can take on only specified, distinct values. A Continuous random variable can take on any value within
More 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 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 informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.2 Transforming and Combining Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers 6.2 Reading Quiz (T or F)
More informationAMS7: WEEK 4. CLASS 3
AMS7: WEEK 4. CLASS 3 Sampling distributions and estimators. Central Limit Theorem Normal Approximation to the Binomial Distribution Friday April 24th, 2015 Sampling distributions and estimators REMEMBER:
More informationChapter 6: Random Variables
Chapter 6: Random Variables Section 6. The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6. 6.3 Binomial and
More informationEXERCISES FOR PRACTICE SESSION 2 OF STAT CAMP
EXERCISES FOR PRACTICE SESSION 2 OF STAT CAMP Note 1: The exercises below that are referenced by chapter number are taken or modified from the following open-source online textbook that was adapted by
More informationModule 4: Probability
Module 4: Probability 1 / 22 Probability concepts in statistical inference Probability is a way of quantifying uncertainty associated with random events and is the basis for statistical inference. Inference
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 informationProbability Theory. Probability and Statistics for Data Science CSE594 - Spring 2016
Probability Theory Probability and Statistics for Data Science CSE594 - Spring 2016 What is Probability? 2 What is Probability? Examples outcome of flipping a coin (seminal example) amount of snowfall
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number
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 information2017 Fall QMS102 Tip Sheet 2
Chapter 5: Basic Probability 2017 Fall QMS102 Tip Sheet 2 (Covering Chapters 5 to 8) EVENTS -- Each possible outcome of a variable is an event, including 3 types. 1. Simple event = Described by a single
More 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 information6.1 Discrete & Continuous Random Variables. Nov 4 6:53 PM. Objectives
6.1 Discrete & Continuous Random Variables examples vocab Objectives Today we will... - Compute probabilities using the probability distribution of a discrete random variable. - Calculate and interpret
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
More informationMATH 446/546 Homework 1:
MATH 446/546 Homework 1: Due September 28th, 216 Please answer the following questions. Students should type there work. 1. At time t, a company has I units of inventory in stock. Customers demand the
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More information7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4 - Would the correlation between x and y in the table above be positive or negative? The correlation is negative. -
More informationChapter 7. Random Variables
Chapter 7 Random Variables Making quantifiable meaning out of categorical data Toss three coins. What does the sample space consist of? HHH, HHT, HTH, HTT, TTT, TTH, THT, THH In statistics, we are most
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 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 informationProbability is the tool used for anticipating what the distribution of data should look like under a given model.
AP Statistics NAME: Exam Review: Strand 3: Anticipating Patterns Date: Block: III. Anticipating Patterns: Exploring random phenomena using probability and simulation (20%-30%) Probability is the tool used
More informationBIOL The Normal Distribution and the Central Limit Theorem
BIOL 300 - The Normal Distribution and the Central Limit Theorem In the first week of the course, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are
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 informationSTAT 201 Chapter 6. Distribution
STAT 201 Chapter 6 Distribution 1 Random Variable We know variable Random Variable: a numerical measurement of the outcome of a random phenomena Capital letter refer to the random variable Lower case letters
More informationNo, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
More informationVIDEO 1. A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled.
Part 1: Probability Distributions VIDEO 1 Name: 11-10 Probability and Binomial Distributions A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled.
More information4.3 Normal distribution
43 Normal distribution Prof Tesler Math 186 Winter 216 Prof Tesler 43 Normal distribution Math 186 / Winter 216 1 / 4 Normal distribution aka Bell curve and Gaussian distribution The normal distribution
More informationChapter 7. Random Variables: 7.1: Discrete and Continuous. Random Variables. 7.2: Means and Variances of. Random Variables
Chapter 7 Random Variables In Chapter 6, we learned that a!random phenomenon" was one that was unpredictable in the short term, but displayed a predictable pattern in the long run. In Statistics, we are
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 information6.1 Discrete and Continuous Random Variables. 6.1A Discrete random Variables, Mean (Expected Value) of a Discrete Random Variable
6.1 Discrete and Continuous Random Variables 6.1A Discrete random Variables, Mean (Expected Value) of a Discrete Random Variable Random variable Takes numerical values that describe the outcomes of some
More information30 Wyner Statistics Fall 2013
30 Wyner Statistics Fall 2013 CHAPTER FIVE: DISCRETE PROBABILITY DISTRIBUTIONS Summary, Terms, and Objectives A probability distribution shows the likelihood of each possible outcome. This chapter deals
More informationBinomial Random Variables
Models for Counts Solutions COR1-GB.1305 Statistics and Data Analysis Binomial Random Variables 1. A certain coin has a 25% of landing heads, and a 75% chance of landing tails. (a) If you flip the coin
More informationChapter 6: Random Variables
Chapter 6: Random Variables Section 6.1 Discrete and Continuous Random Variables The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous
More informationSampling Distributions Chapter 18
Sampling Distributions Chapter 18 Parameter vs Statistic Example: Identify the population, the parameter, the sample, and the statistic in the given settings. a) The Gallup Poll asked a random sample of
More informationShifting our focus. We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why?
Probability Introduction Shifting our focus We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why? What is Probability? Probability is used
More information= 0.35 (or ˆp = We have 20 independent trials, each with probability of success (heads) equal to 0.5, so X has a B(20, 0.5) distribution.
Chapter 5 Solutions 51 (a) n = 1500 (the sample size) (b) The Yes count seems like the most reasonable choice, but either count is defensible (c) X = 525 (or X = 975) (d) ˆp = 525 1500 = 035 (or ˆp = 975
More informationReview of the Topics for Midterm I
Review of the Topics for Midterm I STA 100 Lecture 9 I. Introduction The objective of statistics is to make inferences about a population based on information contained in a sample. A population is the
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 informationProbability & Sampling The Practice of Statistics 4e Mostly Chpts 5 7
Probability & Sampling The Practice of Statistics 4e Mostly Chpts 5 7 Lew Davidson (Dr.D.) Mallard Creek High School Lewis.Davidson@cms.k12.nc.us 704-786-0470 Probability & Sampling The Practice of Statistics
More informationStatistics and Probability
Statistics and Probability Continuous RVs (Normal); Confidence Intervals Outline Continuous random variables Normal distribution CLT Point estimation Confidence intervals http://www.isrec.isb-sib.ch/~darlene/geneve/
More informationChapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
More informationWhat do you think "Binomial" involves?
Learning Goals: * Define a binomial experiment (Bernoulli Trials). * Applying the binomial formula to solve problems. * Determine the expected value of a Binomial Distribution What do you think "Binomial"
More informationDiscrete Probability Distributions
Page 1 of 6 Discrete Probability Distributions In order to study inferential statistics, we need to combine the concepts from descriptive statistics and probability. This combination makes up the basics
More 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 informationMLLunsford 1. Activity: Central Limit Theorem Theory and Computations
MLLunsford 1 Activity: Central Limit Theorem Theory and Computations Concepts: The Central Limit Theorem; computations using the Central Limit Theorem. Prerequisites: The student should be familiar with
More informationNormal distribution. We say that a random variable X follows the normal distribution if the probability density function of X is given by
Normal distribution The normal distribution is the most important distribution. It describes well the distribution of random variables that arise in practice, such as the heights or weights of people,
More informationChapter 8 Homework Solutions Compiled by Joe Kahlig. speed(x) freq 25 x < x < x < x < x < x < 55 5
H homework problems, C-copyright Joe Kahlig Chapter Solutions, Page Chapter Homework Solutions Compiled by Joe Kahlig. (a) finite discrete (b) infinite discrete (c) continuous (d) finite discrete (e) continuous.
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 informationBut suppose we want to find a particular value for y, at which the probability is, say, 0.90? In other words, we want to figure out the following:
More on distributions, and some miscellaneous topics 1. Reverse lookup and the normal distribution. Up until now, we wanted to find probabilities. For example, the probability a Swedish man has a brain
More informationMATH 264 Problem Homework I
MATH Problem Homework I Due to December 9, 00@:0 PROBLEMS & SOLUTIONS. A student answers a multiple-choice examination question that offers four possible answers. Suppose that the probability that the
More informationSection 7.4 Transforming and Combining Random Variables (DAY 1)
Section 7.4 Learning Objectives (DAY 1) After this section, you should be able to DESCRIBE the effect of performing a linear transformation on a random variable (DAY 1) COMBINE random variables and CALCULATE
More informationSTA Module 3B Discrete Random Variables
STA 2023 Module 3B Discrete Random Variables Learning Objectives Upon completing this module, you should be able to 1. Determine the probability distribution of a discrete random variable. 2. Construct
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 information4.1 Probability Distributions
Probability and Statistics Mrs. Leahy Chapter 4: Discrete Probability Distribution ALWAYS KEEP IN MIND: The Probability of an event is ALWAYS between: and!!!! 4.1 Probability Distributions Random Variables
More informationWe use probability distributions to represent the distribution of a discrete random variable.
Now we focus on discrete random variables. We will look at these in general, including calculating the mean and standard deviation. Then we will look more in depth at binomial random variables which are
More informationSection M Discrete Probability Distribution
Section M Discrete Probability Distribution A random variable is a numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are typically denoted
More informationChapter 8. Variables. Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 Random Variables Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc. 8.1 What is a Random Variable? Random Variable: assigns a number to each outcome of a random circumstance, or,
More informationFall 2015 Math 141:505 Exam 3 Form A
Fall 205 Math 4:505 Exam 3 Form A Last Name: First Name: Exam Seat #: UIN: On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work Signature: INSTRUCTIONS Part
More informationLet X be the number that comes up on the next roll of the die.
Chapter 6 - Discrete Probability Distributions 6.1 Random Variables Introduction If we roll a fair die, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6, and each of these numbers has probability
More informationSec$on 6.1: Discrete and Con.nuous Random Variables. Tuesday, November 14 th, 2017
Sec$on 6.1: Discrete and Con.nuous Random Variables Tuesday, November 14 th, 2017 Discrete and Continuous Random Variables Learning Objectives After this section, you should be able to: ü COMPUTE probabilities
More informationSTA Rev. F Learning Objectives. What is a Random Variable? Module 5 Discrete Random Variables
STA 2023 Module 5 Discrete Random Variables Learning Objectives Upon completing this module, you should be able to: 1. Determine the probability distribution of a discrete random variable. 2. Construct
More information4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
More informationExam II Math 1342 Capters 3-5 HCCS. Name
Exam II Math 1342 Capters 3-5 HCCS Name Date Provide an appropriate response. 1) A single six-sided die is rolled. Find the probability of rolling a number less than 3. A) 0.5 B) 0.1 C) 0.25 D 0.333 1)
More information