When you have completed this chapter, you will be able to: Define the terms probability distribution and random variable; distinguish between discrete and continuous random variables; calculate the mean, variance, and standard deviation of a discrete probability distribution; describe the characteristics and compute probabilities using the Poisson probability distribution.
6 1 Discrete Probability Distributions Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 2 When you have completed this chapter, you will be able to: Define the terms probability distribution and Distinguish between discrete and Calculate the mean, variance, and standard deviation of Describe the characteristics and compute probabilities random variable. continuous random variables a discrete probability distribution using the Poisson probability distribution Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Terminology Random Variable …is a numerical value determined by …is a numerical value determined by the outcome of an experiment the outcome of an experiment Probability Distribution …is the listing of all possible outcomes …is the listing of all possible outcomes of an experiment of an experiment and the corresponding probability and the corresponding probability Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 3 Types of Probability Distributions 6 4 Discrete Discrete Under this distribution Under this distribution the random variable random variable the has a has a countable number countable number of possible outcomes of possible outcomes Continuous ontinuous C Under this distribution Under this distribution the random variable random variable the has an has an infinite number infinite number of possible outcomes of possible outcomes Examples Examples Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Types of Probability Distributions 6 5 Discrete Discrete Students in a class Examples Examples Continuous ontinuous C Distance driven by an executive to get to work Number of children in a family Mortgage Loan Number of Mortgages approved in a month Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. The length of time of a particular phone call The length of time of an afternoon nap! Distinguishing featuress Distinguishing feature 6 6 of a of a Discrete Distribution: Distribution: Discrete The sum of the probabilities of the various outcomes is 1.00 The probability of a particular outcome is between 0 and 1.00 The outcomes are mutually exclusive Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 7 Consider a random experiment in which Consider a random experiment in which a coin is tossed three times a coin is tossed three times Let x be the number of Heads Let H represent the outcome of a Let T represent the outcome of Head Tails Determine the probability distribution Determine the probability distribution Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Listing the possibilities Listing the possibilities 6 8 Heads Heads Heads Heads Heads Tails Heads Tails Heads Tails Heads Heads Heads Tails Tails Tails Heads Tails Tails Tails Heads Tails Tails Tails … the possible values of x … the possible values of x (number of heads) are 0,1,2,3 (number of heads) are 0,1,2,3 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 9 Probability Distribution Probability Distribution Consider a Consider a random random experiment in experiment in which a coin is which a coin is tossed three times. tossed three times. Determine the Determine the probability probability distribution distribution What is the probability of tossing 2 heads in 3 flips? P(x) # of Outcomes 1 3/8 3/8 1/8 8/8 = 1 x Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 1/8 Mean of a Discrete Mean of a Discrete Probability Distribution Probability Distribution 6 10 reports the central location of the data is denoted by the Greek symbol , mu is the longrun average value of the random variable also referred to as its expected value, E(X), in a probability distribution is a weighted average Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Binomial Probability Binomial Probability Distribution Distribution 6 25 n = 10 From text Appendix A Probability X 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.599 0.349 0.107 0.028 0.006 0.001 0.000 0.000 0.000 0.000 0.000 0.315 0.387 0.268 0.121 Where represents the 0.040 0.010 0.002 0.000 0.000 0.000 0.000 0.075 0.194 0.302 0.233 0.121 0.044 0.011 0.000 0.000 0.000 0.000 Represents the 0.010 0.057 0.201 0.267 0.215 0.117 0.042 0.009 0.001 0.000 0.000 0.001 0.011 0.088 0.200 0.251 0.205 0.111 0.037 0.006 0.000 0.000 0.000 0.001 0.026 0.103 0.201 0.246 0.201 0.103 0.026 0.001 0.000 0.000 0.000 0.006 0.037 0.111 0.205 0.251 0.200 0.088 0.011 0.001 0.000 0.000 0.001 0.009 0.042 0.117 0.215 0.267 0.201 0.057 0.010 0.000 0.001 0.000 0.001 0.011 0.044 0.121 0.233 0.302 0.194 0.075 Represents the ‘ PROBABILITY’ when 0.000 0.000 0.000 0.000 0.002 0.010 0.040 0.121 0.268 0.387 0.315 10 X ‘Number Unemployed’ ‘Probability’ Explanations Explanations n = 10 0.000 0.001 0.000 0.000 0.000 0.001 0.006 0.028 0.107 0.349 0.599 Using Appendix A Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Using Appendix A Alternate Solution… Alternate Solution… Using Appendix A Using Appendix A Binomial Probability Distribution Binomial Probability Distribution DATA: 20% Unemployed & Sample of 10 Exactly three are unemployed X 0.05 0.10 0.20 0.30 …0.80 0.90 0.95 0.599 0.349 0.107 0.028 …0.000 0.000 0.000 0.315 0.387 0.268 0.121 …0.000 0.000 0.000 0.075 0.194 0.302 0.233 …0.000 0.000 0.000 0.010 0.057 0.201 0.201 0.267 …0.001 0.000 0.000 0.001 0.011 0.088 0.200 …0.006 0.000 0.000 201 or 20.1% Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 26 Alternate Solution… Alternate Solution… Using Appendix A Using Appendix A Binomial Probability Distribution Binomial Probability Distribution DATA: 20% Unemployed & Sample of 10 At least three are unemployed Alternate Reasoning: If ‘at least three are unemployed’ it follows that You can turn this into a problem of ‘at most seven are employed!’ 80% employment if you wish Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 27 Alternate Solution… Alternate Solution… Using Appendix A Using Appendix A Binomial Probability Distribution Binomial Probability Distribution DATA: 20% Unemployed & Sample of 10 X 6 28 At least three are unemployed 0.05 0.10 0.20 To account for the ‘at least 3 unemployed’, 0.599 0.349 0.107 we must TOTAL the percentages from 3 to 10, inclusively 10 0.315 0.387 0.268 0.201 0.088 0.075 0.194 0.302 0.026 0.006 0.010 0.057 0.201 0.001 0.000 0.001 0.011 0.088 0.000 0.000 0.000 0.001 0.026 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 0.322 or 32.2% Alternate Solution… Alternate Solution… Using Appendix A Using Appendix A Binomial Probability Distribution Binomial Probability Distribution DATA: 20% Unemployed & Sample of 10 None are unemployed X 0.05 0.10 0.20 0.30 …0.80 0.90 0.95 0.599 0.349 0.107 0.028 …0.000 0.000 0.000 0.315 0.387 0.268 0.121 …0.000 0.000 0.000 0.075 0.194 0.302 0.233 …0.000 0.000 0.000 0.010 0.057 0.201 0.267 …0.001 0.000 0.000 0.001 0.011 0.088 0.200 …0.006 0.000 0.000 107 or 10.7% Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 29 Alternate Solution… Alternate Solution… 6 30 Using Appendix A Using Appendix A Binomial Probability Distribution Binomial Probability Distribution DATA: 20% Unemployed & Sample of 10 X 0.05 0.10 0.20 10 0.599 0.349 0.107 0.315 0.387 0.268 0.075 0.194 0.302 0.010 0.057 0.201 0.001 0.011 0.088 0.000 0.001 0.026 0.000 0.000 0.006 0.000 0.000 0.001 0.268 At least one is unemployed 0.302 …at most nine …at most nine 0.201 are employed! are employed! 0.088 0.026 = .892 or 89.2% 89.2% 0.006 0.001 0.000 0.000 0.000 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 2 Mean and Variance Mean and Variance of a of a Binomial Probability Distribution Binomial Probability Distribution Formula Formula 6 31 np The Ontario The Ontario Department of 10(.20) Department of == 10(.20) Labour reports that Labour reports that = 2.0 2.0 = 20% of the workforce 20% of the workforce aged between aged between Formula 15 and 19 years Formula 15 and 19 years is unemployed. is unemployed. 10(.20)(.80) == 10(.20)(.80) From a sample of 10 1.60 == 1.60 workers in this age Therefore, the group, calculate: Therefore, the standard deviation is 1.6 = 1.3 standard deviation is 1.6 = 1.3 and np(1 p) Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 32 Poisson Probability Poisson Probability Distribution Distribution Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Poisson Probability Poisson Probability Distribution Distribution 6 33 e Sk The Binomial Distribution becomes more skewed to the right d we ig R Positive Positive ht as the Probability of success become smaller The limiting form of the Binomial Distribution where the probability of success p is small and n is large is called the Poisson Probability Distribution Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Poisson Probability Poisson Probability Distribution Distribution 6 34 Poisson Probability Distribution can be described mathematically using the formula: P( x) Where… µµ ee xx x e x! u is the mean number of successes in a particular interval of time is the constant 2.71828 is the number of successes Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Poisson Probability Poisson Probability Distribution Distribution 6 35 The mean number of successes… can be determined in binomial situations by… np where n is the number of trials and p the probability of a success The variance of the Poisson distribution The variance of the Poisson distribution is also equal to np np is also equal to Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Poisson Probability Poisson Probability Distribution Distribution 6 36 The Daily (Statistics Canada) reports that The Daily (Statistics Canada) reports that 5% of college and bachelor degree students 5% of college and bachelor degree students default on their student loan default on their student loan within 2 years of graduation. within 2 years of graduation. From a sample of 10 students, find the probability that exactly 1 will default on their loan Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Poisson Probability Poisson Probability Distribution Distribution Formula Formula P( x) The Daily The Daily (Statistics Canada) (Statistics Canada) reports that reports that 5% of college 5% of college and bachelor and bachelor degree students degree students default on their default on their student loan within student loan within 2 years of 2 years of Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. P ( 1) = 6 37 x e x! 1 05 e 1 ! u 05 .0476 or 4.76% or 4.76% .0476 Test your learning… … Test your learning … … n o n o k ilcick CCl www.mcgrawhill.ca/college/lind Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s EStat data …and much more! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 38 6 39 This completes Chapter 6 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ... and the corresponding probability and the corresponding probability Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 3 Types of Probability Distributions 6 4 Discrete Discrete... of houses painted 20 of houses painted 20/20 = 1.0 per week: per week: Determine the Probability distribution and its mean and variance Determine the Probability distribution and its mean and variance... Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 6 9 Probability Distribution Probability Distribution Consider a Consider a random random experiment in experiment in which a coin is which a coin is