1 Chapter 6 Discrete Probability Distributions

2 Goals 1.Define the terms: Probability distribution Random variable Continuous probability distributions (Chapter 7) Discrete probability distributions (Chapter 6) 2.For a discrete probability distribution, calculate: Mean Variance & standard deviation 3.Binomial probability distribution: Describe the characteristics Compute probabilities using tables in appendix A Mean of binomial distribution Standard deviation of binomial distribution

3 So Far, & The Future… Chapter 2-4  Descriptive statistics about something that has already happened:  Frequency distributions  Charts  Measures of central tendency (measures of location)  Spread of data: dispersion Starting with Chapter 5, 6  Inferential statistics about something that would probably happen:  Probability rules for:  Addition  Multiplication  Probability distributions:  Gives the entire range of values that can occur based on an experiment  Describes how likely some future event is

4 Define The Terms: Probability distribution Random variable Discrete probability distributions Continuous probability distributions

5 Define: Probability Distribution  A listing of all the outcomes of an experiment and the probability associated with each outcome

6 Random Variable  Random Variable  A quantity resulting from an experiment that, by chance, can assume different values  Because the quantitative or qualitative results from an experiment are due to chance (the future is unknown), we call the variable a random variable  Examples:  Number of students absent any one day  The number of people in class that say “blue” is their favorite color  The number of smiles on any given day  The GDP for a country for any given month  The Russell 500 value at any given second

7 Continuous Probability Distributions (Chapter 7)  A listing of all the outcomes of an experiment with a continuous random variable and the probability associated with each outcome  Continuous Random Variable  A variable that can assume any value within a range  Depending on the measuring instrument

8 Examples Of A Continuous Probability Distribution:  The distance students travel to class  The time it takes an executive to drive to work  The length of an afternoon nap  The length of time of a particular phone call  Weight, height, air pressure, distance  Although, money could be Discrete, it is considered Continuous

9 Discrete Probability Distributions (Chapter 6)  A listing of all the outcomes of an experiment with a discrete random variable and the probability associated with each outcome  Discrete Random Variable  A variable that can assume only certain clearly separated values  Fractions and decimals are o.k.  Discrete random variable is usually the result of counting

10 Examples Of A Discrete Probability Distribution:  The number of students in a class  The number of children in a family  The number of cars entering a carwash in a hour  Number of home mortgages approved by Coastal Federal Bank last week  Number of people at a boomerang tournament  Scores for a dancer at the Olympics

11 Main Features Of A Discrete Probability Distribution:  The probability of a particular outcome is between 0 and 1.00  0 ≤ P(x) ≤ 1  The outcomes are mutually exclusive  The sum of the probabilities of all the mutually exclusive outcomes = 1.00  ΣP(x) = 1

12 Example Of Discrete Probability Distribution The event {two heads} occurs and the random variable = 2

13 Example Of Discrete Probability Distribution The outcome of zero heads occurred once The outcome of one head occurred three times The outcome of two heads occurred three times The outcome of three heads occurred once

14 For Last Experiment (Toss Coin 3 Times)  Toss 1 = trial 1 = 2 possible outcomes  Toss 2 = trial 2 = 2 possible outcomes  Toss 3 = trial 3 = 2 possible outcomes  Sample space = 2 * 2 * 2 = 2^3 = 8

15 Mean & Standard Deviation For A Discrete Probability Distribution Mean (measure of central location) Mean for probability distribution = μ (mu) Standard Deviation (spread of data) Standard deviation for probability distribution = σ (sigma)

16 Mean For Probability Distribution = μ (mu) Reports the central location of the data Is the long-run average value of the random variable Is also referred to as its expected value, E(x) Is a weighted average Possible values are weighted by corresponding probabilities

17 Standard deviation for probability distribution = σ (sigma) Reports the amount of spread (variation) in the distribution Steps: 1.Subtract the mean from each value, and square the difference 2.Multiply each squared difference by its probability 3.Sum the resultant products 4.Take the square root of the result

18 For Probability Distribution Calculate: μ & σ  Tom Jones sells cars for Feeling Good Ford  Tom sells the most cars on Saturdays  The Following Discrete Probability Distribution shows the number of cars he expects to sell on a particular Saturday:  Tom expects to sell only within a certain range of cars (Not 5, 6, 40, or 1/2 car!)"  Outcomes are mutually exclusive

19 For Probability Distribution Calculate: μ

20 For Probability Distribution Calculate: σ

21 For Probability Distribution Calculate: μ  2.1?  Over a large number of Saturdays, Tom expects to sell a mean of 2.1 cars  We cannot sell 2.1 cars  This is why we call the value the “expected value”

22 For Probability Distribution Calculate: σ  Standard deviation = cars?  We can use this to compared to other sales people  If Sue has a mean of 2.1 cars on Saturday but her standard deviation is.93 cars, we can conclude that there is less variability in her Saturday sales that Tom’s Saturday sales  Her values are clustered more closely around the mean

23 Probability Distribution Example 2  Dan Desch, owner of College Painters, studied his records for the past 20 weeks and reports the following number of houses painted per week:

24 Probability Distribution Example 2

25  Compute the mean number of houses painted per week: Probability Distribution Example 2

26  Compute the standard deviation of the number of houses painted per week: Probability Distribution Example 2

27 Binomial Probability Distribution: Describe the characteristics Compute probabilities using tables in appendix A Mean of binomial distribution Standard deviation of binomial distribution

28 The Binomial Distribution Has The Following Characteristics:  It is a discrete probability distribution  Only two possible outcomes for any one trial of a binomial experiment (success or failure)  Question can be true or false  Product can be acceptable or not acceptable  Customer can purchase or not purchase  Citizen can vote or not vote

29 The Binomial Distribution Has The Following Characteristics:  The random variable is the result of counting the number of successes (this will be the horizontal axis)  For 20 flips of the coin, count the number of heads  For 50 boxes of cereal, count the number of boxes that weigh less than the weight printed on the box  The probability of success must be the same for each trial (this is not the probability we will be calculating)  Flip a coin,.5 probability to get a head each trial  Multiple choice test with four answers, ¼ probability to get the answer correct

30 The Binomial Distribution Has The Following Characteristics:  Each trail must be independent of any other trial  The outcome of one trial does not affect the outcome of any other trial  There is no pattern  Example: Multiple choice exam: a,a,a, b,b,b, a,a,a,,b,b,b…

31 For An Experiment To Be Binomial It Must Satisfy The Following Conditions: 1.A random variable (x) counts the # of successes in a fixed number of trials (n)  Trials make up the experiment 2.Each trail must be independent of the previous trial  Outcome of one trial does not affect the outcome of any other trial 3.An outcome on each trial of an experiment is classified into one of two mutually exclusive categories: 1.Success or 2.Failure 4.The probability of success stays the same for each trial (so does the probability of failure) Once these conditions are met, we can build a binomial distribution

32

33 Example Of Building A Binomial Distribution  There are five flights daily from Oakland to Seattle. Suppose the probability that any flight arrives late is.20.  What is the probability that none of the flights are late today? Exactly one flight is late today? etc…  First: Are the tests for a binomial experiment met? 1.Fixed number of trials?  Yes. n = 5 2.Each trial independent?  Statisticians feel comfortable answering yes 3.Success or Failure?  Yes. Late or not late 4.Probability of a success the same for each trial?  Yes. P(S) =.2

34 Example Of Building A Binomial Distribution  Second:  We need the values for P(0), P(1), P(2), P(3), P(4), P(5) for the vertical axis  0, 1, 2, 3, 4, 5 will be used for the horizontal axis

35 We Will Not Use The Formula

36 But We Want To Look At An Example:

37 We will Use Tables In Appendix For Calculating Binomial Probabilities Page 713 Take These Numbers and Plot Them!

38

39 The Table In The Book Will Round, So Use The Formula And Build Your Own Tables In Excel

40 Example 2 Of Building A Binomial Distribution  The West Seattle Bridge is clogged with traffic during the morning commute 10% of the time. If you plan to travel in the morning commute seven times in the next two weeks.  What is the probability that you won’t get stuck in traffic? Get stuck four times?  First: Are the tests for a binomial experiment met? 1.Fixed number of trials?  Yes. n = 7 2.Each trial independent?  Yes 3.Success or Failure?  Yes. Clogged or not clogged 4.Probability of a success the same for each trial?  Yes. P(Clogged With Traffic) =.1

41  Second:  Let’s find the values for P(0), P(1), P(2), P(3), P(4), P(5), P(6), P(7) for the vertical axis  0, 1, 2, 3, 4, 5, 6, 7 will be used for the horizontal axis Example 2 Of Building A Binomial Distribution

42 We will Use Tables In Appendix For Calculating Binomial Probabilities Page 714 Take These Numbers and Plot Them!

43

44 Graphs: If n Remains The Same, But  Increases

45 Graphs: If n Remains The Same, But  Increases As  approaches.5, the graph becomes symmetrical

46 Graphs: If  Remains The Same, But n Increases

47 Graphs: If  Remains The Same, But n Increases As n gets larger, the graph becomes symmetrical

48 Mean & Standard Deviation of the Binomial Distribution

49 Mean & Standard Deviation of the Binomial Distribution  The West Seattle Bridge is clogged with traffic during the morning commute 10% of the time. If you plan to travel in the morning commute seven times in the next two weeks, what is the mean and standard deviation of this distribution? Over the seven days

50 Cumulative Probability Distribution Simply Add to get P(x<=7)

51 Summarize Chapter 6 1.Define the terms: Probability distribution Random variable Discrete probability distributions Continuous probability distributions 2.For a discrete probability distribution, calculate: Mean Variance & standard deviation 3.Binomial probability distribution: Describe the characteristics Compute probabilities using tables in appendix A Mean binomial distribution Standard deviation of binomial distribution

52 Alternative for standard deviation for probability distribution = σ (sigma)

53 Hypergeometric Probability Distribution  Requirements for a Hypergeometric Experiment? 1.An outcome on each trial of an experiment is classified into 1 of 2 mutually exclusive categories: Success or Failure 2.The random variable is the number of counted successes in a fixed number of trials 3.The trials are not independent (For each new trial, the sample space changes 4.We assume that we sample from a finite population (number of items in population is known) without replacement and n/N > So, the probability of a success changes for each trial  N = Number Of Items In Population  n = Number Of Items In Sample = Number Of Trials  S = Number Of Successes In Population  X = Number Of Successes In Sample

54 Hypergeometric Probability Distributio n  If the selected items are not returned to the population and n/N <0.05, then the Binomial Distribution can be used as a close approximation  In Excel use the HYPGEOMDIST function

55 Poisson Probability Distributions  The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution  The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller  This distribution describes the number of times some event occurs during a specified interval (time, distance, area, or volume)  Examples of use:  Distribution of errors is data entry  Number of scratches in car panels

Poisson Probability Experiment 1.The Random Variable is the number of times (counting) some event occurs during a Defined Interval  Random Variable:  The Random Variable can assume an infinite number of values, however the probability becomes very small after the first few occurrences (successes)  Defined Interval:  The Defined Interval some kind of “Continuum” such as:  Misspelled words per page ( continuum = per page)  Calls per two hour period ( continuum = per two hour period)  Vehicles per day ( continuum = per day)  Goals per game ( continuum = per game)  Lost bags per flight ( continuum = per flight)  Defaults per 30 year mortgage period ( continuum = per 30 year mortgage period )  Interval = Continuum 2.The probability of the event is proportional to the size of the interval (the longer the interval, the larger the probability) 3.The intervals do not overlap and are independent (the number of occurrences in 1 interval does not affect the other intervals 4.When the probability of success is very small and n is large, the Poisson distribution is a limiting form of the binomial probability distribution (“Law of Improbable Events”) 56

Poisson Probability 57 In Excel use the POISSON function

58 Poisson Probability Distributions  Variance = Mu  Always positive skew (most of the successes are in the first few counts (0,1,2,3…)  As mu becomes larger, Poisson becomes more symmetrical  We can calculate Probability with only knowledge of Mu and X.  Although using the Binomial Probability Distribution is still technically correct, we can use the Poisson Probability Distribution to estimate the Binomial Probability Distribution when n is large and pi is small  Why? Because as n gets large and pi gets small, the Poisson and Binomial converge

Poisson Probability Distributions 59

Poisson Probability Distributions 60