ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
|
|
- Piers Austin
- 5 years ago
- Views:
Transcription
1 ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23
2 Normal Density Function f (x) = 1 σ 2π e 1 2 (x µ σ )2 where < x < µ and σ are parameters of this distribution X N(µ, σ 2 ) (Fall 2011) Probability Distributions Lecture 8 Part 2 2 / 23
3 Example: mean 0 and s.d. 1 f (x) = 1 1 2π e 1 2 ( x 0 1 )2 f (x) = 1 1 2π e 0.5x 2 f (x) = e 0.5x2 f (x) = x2 x f(x) (Fall 2011) Probability Distributions Lecture 8 Part 2 3 / 23
4 Example: mean 0 and s.d. 1 x f(x) (Fall 2011) Probability Distributions Lecture 8 Part 2 4 / 23
5 (Fall 2011) Probability Distributions Lecture 8 Part 2 5 / 23
6 (Fall 2011) Probability Distributions Lecture 8 Part 2 6 / 23
7 Finding Normal Probabilities Probability is area under the bell curve Finding the area is tricky even with calculus Can use tables and software to find normal probabilities Algorithm for finding normal probabilities: 1 Standardize 2 Read the table (Fall 2011) Probability Distributions Lecture 8 Part 2 7 / 23
8 Standard Normal: Z N(0,1) If X N(µ, σ), then to get standard normal random variable Z with mean 0 and s.d. 1: z = X µ σ Recall: standardization is a linear transformation: z = X µ σ z = µ }{{} σ a + 1 x }{{} σ b (Fall 2011) Probability Distributions Lecture 8 Part 2 8 / 23
9 (Fall 2011) Probability Distributions Lecture 8 Part 2 9 / 23
10 Finding Probabilities with Standard Normal Table Z (Fall 2011) Probability Distributions Lecture 8 Part 2 10 / 23
11 Finding Probabilities Need to find: P(x 1 < X < x 2 ) We can standardize X and find P(z 1 < Z < z 2 ) from table: P( x 1 µ σ < X µ }{{ σ } Z < x 2 µ σ ) Answer is exactly the same. Why? (Fall 2011) Probability Distributions Lecture 8 Part 2 11 / 23
12 Finding Range Probabilities (Fall 2011) Probability Distributions Lecture 8 Part 2 12 / 23
13 Finding Probabilities Here is the intuition: X - the percentage return on investment - is distributed normally with a mean of 10 percent and a s.d. of 10 percent, or X N(10, 10 2 ). What is the probability that return will be greater than 20 percent, or X > 20? X > 20 = X µ > 20 µ = X µ σ > 20 µ σ Therefore, P(X > 20) = P X µ σ }{{} Z > 20 µ σ (Fall 2011) Probability Distributions Lecture 8 Part 2 13 / 23
14 Example The amount of time devoted to studying statistics each week by students is normally distributed with a mean of 7.5 hours and a s.d. of 2.1 hours. What proportion of students study for more than 10 hours per week: P(X > 10) =? P(X > 10) = P(Z > ) = P(Z > 1.19) = Probability that a randomly selected student spends between 7 and 9 hours studying: P(7 < X < 9) =? P(7 < X < 9) = P( < Z < ) = P( 0.24 < Z < 0.71) = P(Z < 0.71) P(Z < 0.24) = (Fall 2011) Probability Distributions Lecture 8 Part 2 14 / 23
15 Find z A such that P (Z> z A ) = A (Fall 2011) Probability Distributions Lecture 8 Part 2 15 / 23
16 Example Cont d The amount of time spent studying X N(7.5, ) If a student is in the top 5%, what amount of time does he/she spend studying? P(X > x A ) = Need to find x A =? P(X > x A ) = 0.05 P(Z > x A ) = 0.05 P(Z > z A ) = 0.05 From table z 0.05 = Un-standardize z A to get back x A x A = x A = = hours (Fall 2011) Probability Distributions Lecture 8 Part 2 16 / 23
17 Z A = X A µ σ X A = Z A σ + µ (Fall 2011) Probability Distributions Lecture 8 Part 2 17 / 23
18 Symmetry of Normal Distribution P(Z < z A ) = P(Z > z A ) (Fall 2011) Probability Distributions Lecture 8 Part 2 18 / 23
19 Summary: Normal Distribution Normal distribution is symmetric and bell-shaped Values are clustered around the mean. Recall Empirical Rule: about 68.3% within 1 s.d. of mean about 95.4% within 2 s.d. of mean about 99.7% within 3 s.d. of mean Parameters: µ and σ Unbounded support: (, ) (Fall 2011) Probability Distributions Lecture 8 Part 2 19 / 23
20 Normal Approximation to Binomial Distribution X B(100, 0.2) P(X = 20) = C (0.2)20 (0.8) 80 = 100! 20!80! (0.2)20 (0.8) 80 = = X N(20, 16) P(X = 20) = P(19.5 < X < 20.5) = P( < Z < 0.125) = (Fall 2011) Probability Distributions Lecture 8 Part 2 20 / 23
21 Rule of Thumb for Binomial Distribution To determine if Normal distribution is a good approximation for the Binomial: Check if the entire interval lies between 0 and n, where interval is given by: np ± 3 np(1 p) What is the concept behind the rule of thumb for Binomial distribution? (Fall 2011) Probability Distributions Lecture 8 Part 2 21 / 23
22 Alternative Rule of Thumb At least two alternative rules of thumb for approximation 1 np > 5 and n(1 p) > 5 2 np > 10 and n(1 p) > 10 3 Obviously, if 2 holds, then 1 holds Use the one you find more intuitive and convenient (Fall 2011) Probability Distributions Lecture 8 Part 2 22 / 23
23 Use Normal Approximation? 10 ± 3 3 (1, 19) Within (0,100) 5 ± ( 1.6, 11.6) Not within (0,100) (Fall 2011) Probability Distributions Lecture 8 Part 2 23 / 23
Lecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
More informationNormal Distribution. Notes. Normal Distribution. Standard Normal. Sums of Normal Random Variables. Normal. approximation of Binomial.
Lecture 21,22, 23 Text: A Course in Probability by Weiss 8.5 STAT 225 Introduction to Probability Models March 31, 2014 Standard Sums of Whitney Huang Purdue University 21,22, 23.1 Agenda 1 2 Standard
More informationLecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
More informationContinuous Probability Distributions & Normal Distribution
Mathematical Methods Units 3/4 Student Learning Plan Continuous Probability Distributions & Normal Distribution 7 lessons Notes: Students need practice in recognising whether a problem involves a discrete
More informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationECO220Y Sampling Distributions of Sample Statistics: Sample Proportion Readings: Chapter 10, section
ECO220Y Sampling Distributions of Sample Statistics: Sample Proportion Readings: Chapter 10, section 10.1-10.3 Fall 2011 Lecture 9 (Fall 2011) Sampling Distributions Lecture 9 1 / 15 Sampling Distributions
More informationSection 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
More informationLecture 12. Some Useful Continuous Distributions. The most important continuous probability distribution in entire field of statistics.
ENM 207 Lecture 12 Some Useful Continuous Distributions Normal Distribution The most important continuous probability distribution in entire field of statistics. Its graph, called the normal curve, is
More informationMath 14 Lecture Notes Ch The Normal Approximation to the Binomial Distribution. P (X ) = nc X p X q n X =
6.4 The Normal Approximation to the Binomial Distribution Recall from section 6.4 that g A binomial experiment is a experiment that satisfies the following four requirements: 1. Each trial can have only
More informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More informationThe graph of a normal curve is symmetric with respect to the line x = µ, and has points of
Stat 400, section 4.3 Normal Random Variables notes prepared by Tim Pilachowski Another often-useful probability density function is the normal density function, which graphs as the familiar bell-shaped
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
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 informationMidTerm 1) Find the following (round off to one decimal place):
MidTerm 1) 68 49 21 55 57 61 70 42 59 50 66 99 Find the following (round off to one decimal place): Mean = 58:083, round off to 58.1 Median = 58 Range = max min = 99 21 = 78 St. Deviation = s = 8:535,
More informationNORMAL RANDOM VARIABLES (Normal or gaussian distribution)
NORMAL RANDOM VARIABLES (Normal or gaussian distribution) Many variables, as pregnancy lengths, foot sizes etc.. exhibit a normal distribution. The shape of the distribution is a symmetric bell shape.
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
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 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 informationAMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao The binomial: mean and variance Recall that the number of successes out of n, denoted
More informationThe Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
More informationTopic 6 - Continuous Distributions I. Discrete RVs. Probability Density. Continuous RVs. Background Reading. Recall the discrete distributions
Topic 6 - Continuous Distributions I Discrete RVs Recall the discrete distributions STAT 511 Professor Bruce Craig Binomial - X= number of successes (x =, 1,...,n) Geometric - X= number of trials (x =,...)
More informationWhat was in the last lecture?
What was in the last lecture? Normal distribution A continuous rv with bell-shaped density curve The pdf is given by f(x) = 1 2πσ e (x µ)2 2σ 2, < x < If X N(µ, σ 2 ), E(X) = µ and V (X) = σ 2 Standard
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 informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More information8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions 8-1
8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions For Example: On August 8, 2011, the Dow dropped 634.8 points, sending shock waves through the financial community.
More informationChapter 7 1. Random Variables
Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous
More informationHomework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a Announcements: There are some office hour changes for Nov 5, 8, 9 on website Week 5 quiz begins after class today and ends at
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 information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial
More informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
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 informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
More 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 informationWeek 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
More information. 13. The maximum error (margin of error) of the estimate for μ (based on known σ) is:
Statistics Sample Exam 3 Solution Chapters 6 & 7: Normal Probability Distributions & Estimates 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the
More informationMATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION
MATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION We have examined discrete random variables, those random variables for which we can list the possible values. We will now look at continuous random variables.
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
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 informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 6 The Normal Distribution 2 Objectives Identify distributions as symmetrical or skewed. Identify the properties of the normal distribution. Find
More information2. The sum of all the probabilities in the sample space must add up to 1
Continuous Random Variables and Continuous Probability Distributions Continuous Random Variable: A variable X that can take values on an interval; key feature remember is that the values of the variable
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 informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 6 Normal Probability Distributions 6-1 Overview 6-2 The Standard Normal Distribution
More informationIntroduction to Business Statistics QM 120 Chapter 6
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can
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 informationSection Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
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 informationSTA258H5. Al Nosedal and Alison Weir. Winter Al Nosedal and Alison Weir STA258H5 Winter / 41
STA258H5 Al Nosedal and Alison Weir Winter 2017 Al Nosedal and Alison Weir STA258H5 Winter 2017 1 / 41 NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION. Al Nosedal and Alison Weir STA258H5 Winter 2017
More informationClass 12. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 12 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 2017 by D.B. Rowe 1 Agenda: Recap Chapter 6.1-6.2 Lecture Chapter 6.3-6.5 Problem Solving Session. 2
More informationDensity curves. (James Madison University) February 4, / 20
Density curves Figure 6.2 p 230. A density curve is always on or above the horizontal axis, and has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. Example
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 Fall 216 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / Fall 216 1
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 27 Continuous
More informationDepartment of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.
Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationProb and Stats, Nov 7
Prob and Stats, Nov 7 The Standard Normal Distribution Book Sections: 7.1, 7.2 Essential Questions: What is the standard normal distribution, how is it related to all other normal distributions, and how
More informationStandard Normal, Inverse Normal and Sampling Distributions
Standard Normal, Inverse Normal and Sampling Distributions Section 5.5 & 6.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 9-3339 Cathy
More informationChapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 9. Introduction This chapter connects the material in Chapters 4 through 8 (numerical descriptive statistics, sampling, and probability distributions, in particular) with
More informationChapter Seven. The Normal Distribution
Chapter Seven The Normal Distribution 7-1 Introduction Many continuous variables have distributions that are bellshaped and are called approximately normally distributed variables, such as the heights
More informationHomework: Due Wed, Feb 20 th. Chapter 8, # 60a + 62a (count together as 1), 74, 82
Announcements: Week 5 quiz begins at 4pm today and ends at 3pm on Wed If you take more than 20 minutes to complete your quiz, you will only receive partial credit. (It doesn t cut you off.) Today: Sections
More informationLecture 5 - Continuous Distributions
Lecture 5 - Continuous Distributions Statistics 102 Colin Rundel January 30, 2013 Announcements Announcements HW1 and Lab 1 have been graded and your scores are posted in Gradebook on Sakai (it is good
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationContinuous random variables
Continuous random variables probability density function (f(x)) the probability distribution function of a continuous random variable (analogous to the probability mass function for a discrete random variable),
More informationChapter ! Bell Shaped
Chapter 6 6-1 Business Statistics: A First Course 5 th Edition Chapter 7 Continuous Probability Distributions Learning Objectives In this chapter, you learn:! To compute probabilities from the normal distribution!
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 informationProbability Distributions II
Probability Distributions II Summer 2017 Summer Institutes 63 Multinomial Distribution - Motivation Suppose we modified assumption (1) of the binomial distribution to allow for more than two outcomes.
More informationcontinuous rv Note for a legitimate pdf, we have f (x) 0 and f (x)dx = 1. For a continuous rv, P(X = c) = c f (x)dx = 0, hence
continuous rv Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a b, P(a X b) = b a f (x)dx.
More information6.2 Normal Distribution. Normal Distributions
6.2 Normal Distribution Normal Distributions 1 Homework Read Sec 6-1, and 6-2. Make sure you have a good feel for the normal curve. Do discussion question p302 2 3 Objective Identify Complete normal model
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 informationMTH 245: Mathematics for Management, Life, and Social Sciences
1/14 MTH 245: Mathematics for Management, Life, and Social Sciences Section 7.6 Section 7.6: The Normal Distribution. 2/14 The Normal Distribution. Figure: Abraham DeMoivre Section 7.6: The Normal Distribution.
More informationUQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions.
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions. Random Variables 2 A random variable X is a numerical (integer, real, complex, vector etc.) summary of the outcome of the random experiment.
More informationIn a binomial experiment of n trials, where p = probability of success and q = probability of failure. mean variance standard deviation
Name In a binomial experiment of n trials, where p = probability of success and q = probability of failure mean variance standard deviation µ = n p σ = n p q σ = n p q Notation X ~ B(n, p) The probability
More informationStatistics 511 Supplemental Materials
Gaussian (or Normal) Random Variable In this section we introduce the Gaussian Random Variable, which is more commonly referred to as the Normal Random Variable. This is a random variable that has a bellshaped
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 28 One more
More informationPart V - Chance Variability
Part V - Chance Variability Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Part V - Chance Variability 1 / 78 Law of Averages In Chapter 13 we discussed the Kerrich coin-tossing experiment.
More informationReview of commonly missed questions on the online quiz. Lecture 7: Random variables] Expected value and standard deviation. Let s bet...
Recap Review of commonly missed questions on the online quiz Lecture 7: ] Statistics 101 Mine Çetinkaya-Rundel OpenIntro quiz 2: questions 4 and 5 September 20, 2011 Statistics 101 (Mine Çetinkaya-Rundel)
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 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 informationStatistical Tables Compiled by Alan J. Terry
Statistical Tables Compiled by Alan J. Terry School of Science and Sport University of the West of Scotland Paisley, Scotland Contents Table 1: Cumulative binomial probabilities Page 1 Table 2: Cumulative
More informationSection 5 3 The Mean and Standard Deviation of a Binomial Distribution!
Section 5 3 The Mean and Standard Deviation of a Binomial Distribution! Previous sections required that you to find the Mean and Standard Deviation of a Binomial Distribution by using the values from a
More informationExamples of continuous probability distributions: The normal and standard normal
Examples of continuous probability distributions: The normal and standard normal The Normal Distribution f(x) Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.
More informationChapter 4 Random Variables & Probability. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random variable =
More informationLecture 8 - Sampling Distributions and the CLT
Lecture 8 - Sampling Distributions and the CLT Statistics 102 Kenneth K. Lopiano September 18, 2013 1 Basics Improvements 2 Variability of Estimates Activity Sampling distributions - via simulation Sampling
More informationData Analysis and Statistical Methods Statistics 651
Review of previous lecture: Why confidence intervals? Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Suppose you want to know the
More informationLecture 6: Normal distribution
Lecture 6: Normal distribution Statistics 101 Mine Çetinkaya-Rundel February 2, 2012 Announcements Announcements HW 1 due now. Due: OQ 2 by Monday morning 8am. Statistics 101 (Mine Çetinkaya-Rundel) L6:
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 informationThe Normal Approximation to the Binomial
Lecture 16 The Normal Approximation to the Binomial We can calculate l binomial i probabilities bbilii using The binomial formula The cumulative binomial tables When n is large, and p is not too close
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
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 informationPROBABILITY DISTRIBUTIONS
CHAPTER 3 PROBABILITY DISTRIBUTIONS Page Contents 3.1 Introduction to Probability Distributions 51 3.2 The Normal Distribution 56 3.3 The Binomial Distribution 60 3.4 The Poisson Distribution 64 Exercise
More informationUnit2: Probabilityanddistributions. 3. Normal and binomial distributions
Announcements Unit2: Probabilityanddistributions 3. Normal and binomial distributions Sta 101 - Summer 2017 Duke University, Department of Statistical Science PS: Explain your reasoning + show your work
More informationMA131 Lecture 9.1. = µ = 25 and σ X P ( 90 < X < 100 ) = = /// σ X
The Central Limit Theorem (CLT): As the sample size n increases, the shape of the distribution of the sample means taken with replacement from the population with mean µ and standard deviation σ will approach
More informationA continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
More informationStandard Normal Calculations
Standard Normal Calculations Section 4.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 10-2311 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationNormal Distribution. Definition A continuous rv X is said to have a normal distribution with. the pdf of X is
Normal Distribution Normal Distribution Definition A continuous rv X is said to have a normal distribution with parameter µ and σ (µ and σ 2 ), where < µ < and σ > 0, if the pdf of X is f (x; µ, σ) = 1
More informationCommonly Used Distributions
Chapter 4: Commonly Used Distributions 1 Introduction Statistical inference involves drawing a sample from a population and analyzing the sample data to learn about the population. We often have some knowledge
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 informationDiploma in Business Administration Part 2. Quantitative Methods. Examiner s Suggested Answers
Cumulative frequency Diploma in Business Administration Part Quantitative Methods Examiner s Suggested Answers Question 1 Cumulative Frequency Curve 1 9 8 7 6 5 4 3 1 5 1 15 5 3 35 4 45 Weeks 1 (b) x f
More informationMATH 3200 Exam 3 Dr. Syring
. Suppose n eligible voters are polled (randomly sampled) from a population of size N. The poll asks voters whether they support or do not support increasing local taxes to fund public parks. Let M be
More information