Continuous Random Variables

Slides:



Advertisements
Similar presentations
The Normal Distribution
Advertisements

Note 7 of 5E Statistics with Economics and Business Applications Chapter 5 The Normal and Other Continuous Probability Distributions Normal Probability.
Continuous Random Variables and Probability Distributions
Chapter Five Continuous Random Variables McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 6 The Normal Distribution and Other Continuous Distributions
Ch. 6 The Normal Distribution
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-1 Chapter 6 The Normal Distribution and Other Continuous Distributions.
Continuous Random Variables and Probability Distributions
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 6-1 Statistics for Managers Using Microsoft® Excel 5th Edition.
CHAPTER 3: The Normal Distributions Lecture PowerPoint Slides The Basic Practice of Statistics 6 th Edition Moore / Notz / Fligner.
Chapter 5: Continuous Random Variables
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 6-1 Chapter 6 The Normal Distribution Business Statistics: A First Course 5 th.
The Normal Distribution
Chapter 6: Normal Probability Distributions
Chapter 13 Statistics © 2008 Pearson Addison-Wesley. All rights reserved.
Chapter 4 Continuous Random Variables and Probability Distributions
4. Random Variables A random variable is a way of recording a quantitative variable of a random experiment.
Chap 6-1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Chapter 6 The Normal Distribution Business Statistics: A First Course 6 th.
Chapter 6 The Normal Probability Distribution
Continuous Probability Distributions  Continuous Random Variable  A random variable whose space (set of possible values) is an entire interval of numbers.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 1 PROBABILITIES FOR CONTINUOUS RANDOM VARIABLES THE NORMAL DISTRIBUTION CHAPTER 8_B.
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-1 Chapter 6 The Normal Distribution and Other Continuous Distributions.
Review A random variable where X can take on a range of values, not just particular ones. Examples: Heights Distance a golfer hits the ball with their.
CONTINUOUS RANDOM VARIABLES AND THE NORMAL DISTRIBUTION.
Probabilistic & Statistical Techniques Eng. Tamer Eshtawi First Semester Eng. Tamer Eshtawi First Semester
Continuous Random Variables Continuous Random Variables Chapter 6.
McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Continuous Random Variables Chapter 6.
Chapter 6 Some Continuous Probability Distributions.
CHAPTER 3: The Normal Distributions ESSENTIAL STATISTICS Second Edition David S. Moore, William I. Notz, and Michael A. Fligner Lecture Presentation.
CHAPTER 3: The Normal Distributions
The Normal Distribution Chapter 6. Outline 6-1Introduction 6-2Properties of a Normal Distribution 6-3The Standard Normal Distribution 6-4Applications.
Introduction to Probability and Statistics Thirteenth Edition
Module 13: Normal Distributions This module focuses on the normal distribution and how to use it. Reviewed 05 May 05/ MODULE 13.
Lesson 2 - R Review of Chapter 2 Describing Location in a Distribution.
Chapter 7 Lesson 7.6 Random Variables and Probability Distributions 7.6: Normal Distributions.
STATISTIC & INFORMATION THEORY (CSNB134) MODULE 7C PROBABILITY DISTRIBUTIONS FOR RANDOM VARIABLES ( NORMAL DISTRIBUTION)
Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 6 Probability Distributions Section 6.2 Probabilities for Bell-Shaped Distributions.
Lecture PowerPoint Slides Basic Practice of Statistics 7 th Edition.
© 2003 Prentice-Hall, Inc. Chap 5-1 Continuous Probability Distributions Continuous Random Variable Values from interval of numbers Absence of gaps Continuous.
© 2002 Prentice-Hall, Inc.Chap 6-1 Basic Business Statistics (8 th Edition) Chapter 6 The Normal Distribution and Other Continuous Distributions.
Continuous Random Variables Continuous random variables can assume the infinitely many values corresponding to real numbers. Examples: lengths, masses.
MATB344 Applied Statistics Chapter 6 The Normal Probability Distribution.
§ 5.3 Normal Distributions: Finding Values. Probability and Normal Distributions If a random variable, x, is normally distributed, you can find the probability.
Chapter 6 The Normal Distribution and Other Continuous Distributions.
INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2011 Pearson Education, Inc. Chapter 16 Continuous Random.
Basic Business Statistics
Review Continuous Random Variables Density Curves
Introduction to Probability and Statistics Thirteenth Edition Chapter 6 The Normal Probability Distribution.
© 2002 Prentice-Hall, Inc.Chap 5-1 Statistics for Managers Using Microsoft Excel 3 rd Edition Chapter 5 The Normal Distribution and Sampling Distributions.
Lesson 2 - R Review of Chapter 2 Describing Location in a Distribution.
1 ES Chapter 3 ~ Normal Probability Distributions.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 6-1 Statistics for Managers Using Microsoft® Excel 5th Edition.
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 6-1 Chapter 6 The Normal Distribution and Other Continuous Distributions Basic Business.
Chap 6-1 Chapter 6 The Normal Distribution Statistics for Managers.
© 2003 Prentice-Hall, Inc. Chap 5-1 Continuous Probability Distributions Continuous Random Variable Values from interval of numbers Absence of gaps Continuous.
Chapter 6 (part 2) WHEN IS A Z-SCORE BIG? NORMAL MODELS A Very Useful Model for Data.
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 6-1 Chapter 6 The Normal Distribution Business Statistics, A First Course 4 th.
Review Continuous Random Variables –Density Curves Uniform Distributions Normal Distributions –Probabilities correspond to areas under the curve. –the.
MATB344 Applied Statistics
Chapter 5: Continuous Random Variables
Distributions Chapter 5
Continuous Random Variables
STAT 206: Chapter 6 Normal Distribution.
Introduction to Probability and Statistics
Statistics for Managers Using Microsoft® Excel 5th Edition
Lecture 12: Normal Distribution
Chapter 5: Continuous Random Variables
Chapter 5: Continuous Random Variables
The Normal Distribution
Presentation transcript:

Continuous Random Variables Chapter 4 Continuous Random Variables Slides for Optional Sections Section 5.2 The Uniform Distribution slide 3 Section 5.6 The Exponential Distribution slides 22-28

Continuous Probability Distributions Continuous Probability Distribution – areas under curve correspond to probabilities for x Area A corresponds to the probability that x lies between a and b Do you see the similarity in shape between the continuous and discrete probability distributions?

The Uniform Distribution Uniform Probability Distribution – distribution resulting when a continuous random variable is evenly distributed over a particular interval Probability Distribution for a Uniform Random Variable x Probability density function: Mean: Standard Deviation:

The Normal Distribution A normal random variable has a probability distribution called a normal distribution The Normal Distribution Bell-shaped curve Symmetrical about its mean μ Spread determined by the value of it’s standard deviation σ

The Normal Distribution The mean and standard deviation affect the flatness and center of the curve, but not the basic shape

The Normal Distribution The function that generates a normal curve is of the form where  = Mean of the normal random variable x  = Standard deviation  = 3.1416… e = 2.71828… P(x<a) is obtained from a table of normal probabilities

The Normal Distribution Probabilities associated with values or ranges of a random variable correspond to areas under the normal curve Calculating probabilities can be simplified by working with a Standard Normal Distribution A Standard Normal Distribution is a Normal distribution with  =0 and  =1 The standard normal random variable is denoted by the symbol z

The Normal Distribution Table for Standard Normal Distribution contains probability for the area between 0 and z Partial table below shows components of table Value of z a combination of column and row Probability associated with a particular z value, in this case z=.13, p(0<z<.13) = .0517

The Normal Distribution What is P(-1.33 < z < 1.33)? Table gives us area A1 Symmetry about the mean tell us that A2 = A1 P(-1.33 < z < 1.33) = P(-1.33 < z < 0) +P(0 < z < 1.33)= A2 + A1 = .4082 + .4082 = .8164

The Normal Distribution What is P(z > 1.64)? Table gives us area A2 Symmetry about the mean tell us that A2 + A1 = .5 P(z > 1.64) = A1 = .5 – A2=.5 - .4495 = .0505

The Normal Distribution What is P(z < .67)? Table gives us area A1 Symmetry about the mean tell us that A2 = .5 P(z < .67) = A1 + A2 = .2486 + .5 = .7486

The Normal Distribution What is P(|z| > 1.96)? Table gives us area .5 - A2 =.4750, so A2 = .0250 Symmetry about the mean tell us that A2 = A1 P(|z| > 1.96) = A1 + A2 = .0250 + .0250 =.05

The Normal Distribution What if values of interest were not normalized? We want to know P (8<x<12), with μ=10 and σ=1.5 Convert to standard normal using P(8<x<12) = P(-1.33<z<1.33) = 2(.4082) = .8164

The Normal Distribution Steps for Finding a Probability Corresponding to a Normal Random Variable Sketch the distribution, locate mean, shade area of interest Convert to standard z values using Add z values to the sketch Use tables to calculate probabilities, making use of symmetry property where necessary

The Normal Distribution Making an Inference How likely is an observation in area A, given an assumed normal distribution with mean of 27 and standard deviation of 3? z value for x=20 is -2.33 P(x<20) = P(z<-2.33) = .5 - .4901 = .0099 You could reasonably conclude that this is a rare event

The Normal Distribution You can also use the table in reverse to find a z-value that corresponds to a particular probability What is the value of z that will be exceeded only 10% of the time? Look in the body of the table for the value closest to .4, and read the corresponding z value z = 1.28

The Normal Distribution Which values of z enclose the middle 95% of the standard normal z values? Using the symmetry property, z0 must correspond with a probability of .475 From the table, we find that z0 and –z0 are 1.96 and -1.96 respectively.

The Normal Distribution Given a normally distributed variable x with mean 100,000 and standard deviation of 10,000, what value of x identifies the top 10% of the distribution? The z value corresponding with .40 is 1.28. Solving for x0 x0 = 100,000 +1.28(10,000) = 100,000 +12,800 = 112,800

Descriptive Methods for Assessing Normality Evaluate the shape from a histogram or stem-and-leaf display Compute intervals about mean and corresponding percentages Compute IQR and divide by standard deviation. Result is roughly 1.3 if normal Use statistical package to evaluate a normal probability plot for the data

Approximating a Binomial Distribution with a Normal Distribution You can use a Normal Distribution as an approximation of a Binomial Distribution for large values of n Often needed given limitation of binomial tables Need to add a correction for continuity, because of the discrete nature of the binomial distribution Correction is to add .5 to x when converting to standard z values Rule of thumb: interval +3 should be within range of binomial random variable (0-n) for normal distribution to be adequate approximation

Approximating a Binomial Distribution with a Normal Distribution Steps Determine n and p for the binomial distribution Calculate the interval Express binomial probability in the form P(x<a) or P(x<b)–P(x<a) Calculate z value for each a, applying continuity correction Sketch normal distribution, locate a’s and use table to solve