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CHAPTER 6 DISCRETE PROBABILITY DISTRIBUTIONS
Objectives:
1. The student will be able to distinguish between a
discrete and continuous random variable.
2. The student will be able to apply and interpret
expected value for discrete variables.
3. The student will be able to use the expected value
concept with decision trees.
4. The student will be able to apply expected value to
the concepts of risk.
5. The student will be able to use and interpret the
binomial probability distribution.
6. The student will be able to use and interpret the
poisson probability distribution.
Random variable - a variable whose numerical value is
determined by the outcome of a random trial Pg 188
Discrete random variable is able to take on a countable number
of values in an interval. Pg 184
Continuous random variable is assumed able to take any value in
an interval. Pg 184
A probability distribution reports all possible outcomes as well
as the corresponding probability. page 184.
Expected value of a discrete random variable is a weighted mean
 equal to the sum of the products of each value x of the
variable and associated probability P(X = x), or Pg 185
𝜇 = ∑ 𝑥𝑃(𝑥)
𝜎 2 = ∑(𝑥 − 𝜇)2 𝑃(𝑥)
AVERAGE SALES/DAY
REL. FREQ.
2
50
150
250
350
.1
.2
.4
.3
_
X = 200
E(X) = 240
σ = 94.33
COMPUTE EXPECTED VALUE AND A SIMPLE AVERAGE
LOTTERY 2,000,000 entries $1 PER TICKET $1,000,000 FIRST
PRIZE
COMPUTE THE EXPECTED VALUE OF THE LOTTERY TICKET.
COMPUTE THE AVERAGE PROFIT(LOSS) OF THE LOTTERY TICKET.
DRAW A DECISION TREE FOR THE POSSIBLE OUTCOMES.
RISK TAKER - AN INDIVIDUAL IS WILLING TO PAY TO TAKE A
RISK.
[Cyber rebate offers a Fisher Price play phone for $206 with a
rebate of $206. Assume the cost for this product to Cyber Rebate
of this product is $10. Cyber Rebate also notes that 10% of its
customers do not apply for the rebate. Can Cyber Rebate make a
profit with this rebate program?] WSJ 3/5/01 Marketplace
11/25/05 “Beyond Bland: Gift Cards Now Play Music, Record
Messages” WSJ B1
10/13/09
Retailers Turn to Gift-Card Promotions to Lure Reluctant Buyers, Boost
Spending WSJ B1
11/17/09
Fed Targets Gift-Card Fees WSJ A2
Tennis balls on the Web.
50 free downloads
How should you decide if is a good idea to give something away.
If the benefits exceed the costs
What is the cost? Assume e-music pays royalties of .05 per song
What are the benefits? Subscribe for 9.95 a month but I can
cancel anytime in the first 2 weeks and keep 50 songs.
Piracy and formula 6-1
11/19/09
Armed U.S. Ship Repels Attack by Somali Pirates WSJ A12
Piracy and Business decision making
10,000 ships pass near Somilia 40 are taken each year.
What is the probability of being hijacked?
3
FIRE INSURANCE
Mr. Askil faces a decision to buy or not buy fire insurance
for his home. The cost of the insurance is $150/year, but there
is only a 1/1000 chance of a fire destroying their home each
year. Draw a decision tree for the outcomes Mr. Askil faces.
RISK AVERSION - An individual must be compensated to take a risk
or is willing to pay something to avoid a risk.
Mr. Askil has a business opportunity to purchase 10 acres of
land at $50,000/acre. The land will only produce a profit if
Mr.
Askil can get approval from the local government to re-zone the
land for residential use. Mr. Askil can buy an option for
$10,000/acre to buy in the next month. Mr. Askil believes there
is a 50% chance he can convince the local government to rezone
the land. If the land is re-zoned, it can be developed for a
profit of $30,000/acre. Mr. Askil may decide to seek approval
from the local government before making any purchase, but he
estimates there will only be a 30% chance the land will still be
available.
[Compute the expected value of each decision.]
[Explain which branch of the decision tree you would choose.]
[Compute the standard deviation of each branch.]
DRAW THE OUTCOME TREE FOR THIS PROBLEM.
Make a decision.
DRAW OUTCOME TREE FOR A MANUFACTURER REBATE.
RISK NEUTRAL - An individual is indifferent between two assets,
regardless of risk, as long as the expected values are the same.
DISCUSS MARCH MADNESS.
DISCUSS REBATES
TORO START P(S) = .9
Attempts
$50 COST/MACHINE
100,000 SOLD THIS SUMMER 1,200,000
4
1. What is the average number of times a Toro will not
start on two pulls this summer?
2. What is the average cost per lawn mower to Toro of this
plan?
Binomial and poisson probability distributions are used to
determine the probability of events that can occur in more than
one way. In the previous chapters we computed probability of
events occurring in a specific manner.
Binomial distribution represents the probabilities of various
numerical outcomes over several identical, independent trials,
where there are two possible outcomes for each trial. Pg 194
ASSUMPTIONS
1. MUTUALLY EXCLUSIVE, ONLY TWO OUTCOMES ARE POSSIBLE
2. PROBABILITY OF SUCCESS AND FAILURE REMAIN CONSTANT
3. TRIALS ARE INDEPENDENT
TOSS A COIN THREE TIMES
OUTCOMES
0
1
2
3
PROBABILITY
.125
.375
.375
.125
BINOMIAL FORMULA
𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 = 𝑛𝐶𝑥 𝜋 𝑥 (1 − 𝜋)𝑛−𝑥
x = number of successes,
failures
n = number of trials, n-x = number of
10/10/06
“More Fliers Forced To Give Up Seats” WSJ D1
9/17/09 Bumped Passengers Learn a Cruel Flying Lesson WSJ D1
1/07/09
An Airline Report Card: Fewer Delays, Hassles Last Year, but Bumpy
Times May Be Ahead WSJ D1
[Askil is a small commuter airline with a capacity of 14
passengers per plane. Askil also determines that 95% of people,
who have reservations, take the flight. On Reserve
A. What percent of the time will this airplane fly with at least
one empty seat? 5 pts
B. The airline decides to over book this flight by one
person(take 15 reservations). What is the probability that 15
people show up for the flight. 5 pts
5
C. The FAA(Federal Aviation Association) requires compensation
if you are bumped. Assume this compensation cost the airline
$100 for each individual that is bumped(can not get on the plane
because more people have tickets than seats on the aircraft). If
this airline consistently over books by one person, determine
the average cost of over booking this flight. Label and
interpret your answer. 10 pts] Fall 01
Excel - top menu - formula paste function fx - binomial
MEAN AND STANDARD DEVIATION
E(X) = n∏
π = Probability of event A
(1-p) = Probability of the complement of event A.
𝜎𝑥 = √𝑛𝜋(1 − 𝜋)
Function key > Probability Distribution>Binomial
In the dialog box select Probability, set the number of trials
to 6, the probability of success to .5 and the input column to
c1
trials (n=3) and the probability of success is .5 (p=.5)
Poisson distribution represents the random arrival of events per
unit of time, distance or area.
 = the mean number of Poisson distributed events over
the sampling medium that is being examined.
x = the number of occurrences over the sampling medium
ASSUMPTIONS
1. All the assumptions relating to the binomial plus
2. A precise maximum does not exist in the sample
space.
Compare the average number of occurrences to the actual number
of
occurrences to determine the probability.
Determine the poisson approximation to the binomial distribution
for tossing a coin 25 times and getting 12 heads.
Excel - fx - poisson
Chapter 7 CONTINUOUS PROBABILITY DISTRIBUTIONS
Objectives:
6
1. The student will be able to use and interpret the
normal distribution.
2. The student will be able to compute and interpret a
value for a continuous variable if given a
probability of occurrence.
3. The student will be able to know when it is
appropriate to approximate the binomial
distribution with the normal distribution.
4. The student will be able to determine when it is
appropriate to use the binomial, poisson, and
normal distributions.
A PROBABILITY DISTRIBUTION IS CONTINUOUS WHEN THE RANDOM
VARIABLE MAY ASSUME ANY VALUE WITHIN SOME SPECIFIED RANGE.
THE NORMAL DISTRIBUTION - Pg 227
PROPERTIES OF THE NORMAL CURVE
1. Only the mean (μ) and standard deviation (σ) need to
be known to compute probabilities for the normal
distribution.
2. The graph of a normal distribution is bell shaped and
symmetrical around the mean.
3. Since the normal curve is measured on a continuous
scale the probability of obtaining a precise value is
approximately 0.
4. The probability that a random variable will have a
value between any two points is equal to the area under
the normal curve between those two points.
5. The area under the normal curve between the mean and
any other point can be determined by knowing how many
standard deviations this data value is from the mean.
𝑍 𝑣𝑎𝑙𝑢𝑒 =
SHOW TORO SNOWTHROWER AD
Excel -
𝑥−𝜇
𝜎
fx - standardize - for Z-values
fx - normdist - Returns the probability of less than a
specified value for a given mean and standard
deviation. For less than probability cumulative
probability put 1 in the cumulative window.
X
7
Mean
S Dev
Cum
fx - normsdist converts z-values to less than
probabilities. Z
fx - normsinv - Gives a Z-value for a given
probability.
fx - norminv - Gives a data value. Only a
designated percent of the data will be less than
or equal to this value given a mean and standard
deviation.
[Your product requires ball bearings that have an average
diameter 1 cm. You are looking for a supplier of ball bearings.
The quality control department determines that a ±.008 cm
variation around the mean is acceptable.
The mean diameter of the ball bearings is 1 cm. What percent of
parts will be acceptable if the standard deviation is .008?
The quality control department wants to establish a failure rate
of .0026 or .26%. What standard deviation should the quality
control department demand from its suppliers?
This is a 3-sigma quality control level.]
[4.5-sigma .49996599x2 = .999993198 1 - .999993198=.000003198]
Incentive Pay
(How would you design a pay incentive plan for cashiers?)
NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION
𝑋 − 𝑛𝜋
𝑍=
√𝑛𝜋(1 − 𝜋)
Rule of thumb np(1-p)  5 Pg 238
1.
2.
3.
4.
5.
at
6.
at
7.
Determine the probability of 3 defectives in a box of 5.
Determine the probability of 5 or more heads in 20 tosses.
Determine the probability of 3 defects in a yard of cloth.
Determine the probability of 4 phone calls in two hours.
Determine the probability of an individual's income, picked
random, being equal to $36,000.
Determine the probability of an individual's income, picked
random, being greater than $36,000.
Determine the probability of 400 heads in 1000 tosses.
8
Continuity correction is an adjustment that we make by adding or
subtracting ½ to a discrete value when we use a continuous
distribution to approximate a discrete distribution. Pg 238
Normal approximation of a binomial proportion of successes.
⏞
𝑝−𝑝
𝑍=
√𝑝(1 − 𝑝)⁄𝑛
Sometimes, an appropriate procedure is to work not with the
number X of successes in n trials but, instead with the
proportion or fraction of successes.
[What is the probability that a data value is more than 1.96
standard deviations from its mean?]
[Assume 1=0, what is the probability that a population slope is
more than 1.96 standard deviations from 0?]
Chapter 8 Sampling Distributions
Objectives:
1. The student will be able to understand the
importance of random sampling in the process of
inference.
2. The student will be able to distinguish between a
population and a sample. Pg 264-265
3. The student will be able to choose the appropriate
sampling method (i.e. systematic, stratified,
cluster or simple random samples). Pg 265-270.
SAMPLING TECHNIQUES
1. sample versus census
why use a sample?
5/8/10
In Counting Illegal Immigrants, Certain Assumptions Apply WSJ A2
2. Sample should be representative of the population random
If samples are random we can use the normal distribution. There
are N!/[n!(N - n)!] samples of size n in a finite population of
9
size, N, and random sampling means choosing one in such a way
that all are equally likely to be choose.
1.
2.
3.
4.
5.
POPULATION
ALL MEN
COLLEGE STUDENTS
VOTERS
UNEMPLOYED
T.V. VIEWERS
9/6/07
1/10/08
SAMPLE
IS THIS RANDOM
MEN IN THIS CLASS
UW-PLATTEVILLE
PHONE SURVEY BEFORE THE ELECTION
PEOPLE WHO RECEIVE U.E. COMP.
5500 VIEWERS RANDOMLY SELECTED
“New Way to Count Listeners Shakes Up Radio” WSJ B1 survey
Thomas E. Obama WSJ A14
Sampling  Computations  Results  Inference
Why Sample? 1. Impractical (destructive testing) 2.Time and Money Pg 264-265
NON-RANDOM SAMPLING
1. JUDGEMENT SAMPLING
2. QUOTA OR CONVENIENCE SAMPLING
RANDOM SAMPLING
I. Complete list of the population is available - random
numbers with minitab.
Calc-random data.
4/8/09
Which Is Epidemic-Sexting or Worrying About It? WSJ A9
sampling
2/27/10
Lights, Camera, Calculator! The New Celebrity Math WSJ A2 Sampling
II. Population list is not available
Non-human subjects
1. Systematic sampling population does not
follow a pattern - Data set homogeneous-Usually
line
non-human -Assembly
Human Subjects
2. Stratified sample from some subgroup - Data set is
not homogenous, such as
human subjects - effects of drugs on people, advertising dollars based on
return on equity
3. Cluster sample- Sampling a geographic area - divide sample into small units
called primary units entire subgroups- Pg 270
10/18/06
“655,00 War Dead? WSJ A20 Cluster sampling
Good samples are ones that are representative of the
population.
In general if the sample is random it should be representative of the population.
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QUESTIONNAIRE CONSTRUCTION
1. NON-RESPONSE BIAS
PROPER STEPS FOR CONSTRUCTING QUESTIONNAIRES
1. DEFINE YOUR OBJECTIVES
2. FORMULATE THE QUESTIONS
3. DETERMINE THE TABULATION METHOD
MAKE SURE YOU CAN QUANTIFY THE DATA
4. PREPARE THE INSTRUMENT
5. PRETEST THE INSTRUMENT
AER papers and Proceedings May 2001 ‘Do People Mean What they Say? Implications for
Subjective Survey Data’ Pg 67
Cognitive problems
1. Ordering of questions
How happy are you with life in general?
How often do you go out on a date
2. Wording Issues
Do you think the United States should forbid public speeches against democracy? Over
50% said yes
Do you think that the Unites States should allow public speeches against democracy?
75% answered no
3. Scale effect
People are asked to indicated the amount of time they spend watching TV. Half of the
people were given the following scale on the left the other half were given the scale on
the right:
Less than ½
½ to 1 hr
1 hr to 1.5
1.5 to 2.0
2.0 to 2.5 2.5 or less
2.5 to 3.0 2.5 to 3.0
3.0 to 3.5 3.0 to 3.5
3.5 to 4.0 3.5 to 4.0
4.0 to 4.5 4.0 to 4.5
4.5 or more 4.5 or more
11
In the first survey 16%of the people survey indicated they watched more than 2.5 hours
of TV per day, while 32% of the people in the second survey indicated they watched
more than 2.5 hours.
Social Desirability
Respondents want to avoid looking bad.
Non-Attitudes, Wrong Attitudes and Soft Attitudes
Respondents reluctance to admit lack of an attitude.
People are asked about things they do not know
Unclear about their attitudes (rope experiment)
Experiment when people are paid a little or a lot.
The probability distribution of a statistic is known as the statistic's sampling distribution of the
sample means. Pg 267
Assignment:
One fourth of the groups in the class ask the following question to students in your other classes:
Do you favor or oppose allowing students and parents to choose a private school to
attend at tax payer expense?
One fourth of the groups in the class ask the following question to students in your other classes:
Do you favor or oppose that tax dollars should be used to assist parents who send their children
to private, parochial or religious schools, or should tax dollars be spent to improve public
schools?
One fourth of the groups in the class ask the following question to students in your other classes:
Do you favor or oppose allowing students and parents to choose a private school to
attend, assuming this policy will result in tax increases?
One fourth of the groups in the class ask the following question to students in your other classes:
Do you favor or oppose allowing students and parents to choose a private school to
attend, assuming this policy will result in no increase in taxes?
Groups with the same question should count the total number sampled and present the
percentage of people in favor or opposed.
[Wis State Journal 11/10/08 A3
A advisory referendum was placed on the ballot of 11/4/08
“Shall the next state Legislature enact health-care reform by December 31, 2009, that guarantees
every Wisconsin resident affordable health-care coverage as good as what is provided to state
legislators?”
12/1/10
Race Is On to 'Fingerprint' Phones, PCs WSJ A1
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Would the phrasing of this question have any impact on the final results of the referendum?]
REFERENCES:
3/ /85 “...to highest bidders” WSJ
3/24/89 “Hottest Commodity In Wall Street Pits? Georgetown
Hoyas” WSJ
FALL 1990 "ON THE ECONOMICS OF STATE LOTTERIES" by Charles T.
Clotfelter and Philip J. Cook J.E.P.
Spring 1990 “Quality...World Class Definition” Applied Microwave
magazine
7/19/90 “T.V. Neilson Ratings Long Unquestioned, Face Tough
Challenges” WSJ
3/18/91
“Marketers Zero In On Their Customers” WSJ
11/14/91 “Studies Galore Support Products and Positions, But
Are They
Reliable? WSJ
7/28/93
“Statisticians Occupy Front Lines In Battle Over
Passive
Smoking” WSJ
5/30/95
“Lottery takes biggest bite from wallets of poor” Wis.
State Journal
1/21/96 “In all probability, polls will never reach perfection”
WSJ
4/11/96
“Fright by the Numbers: Alarming Disease Data Are
Frequently
Flawed” WSJ
5/2/96 “Preparing for 2000, Census Bureau Tests Carrots vs.
Sticks” WSJ
8/27/96
“Polling Quirks Give HMOs Healthy Ratings” WSJ
9/17/96
“What Americans Really think of School Choice” WSJ 11/22/96 “Networks
Blast Nielson, Blame Faulty Ratings for Drop
in Viewership” WSJ
8/30/97
“Tempest in a Census” WSJ
10/16/97 “No Easy money: Powerball lottery game will be harder to
win” Wis.
State Journal
2/10/98
“Rebates Secret Appeal to Manufacturers: Few Consumers
Actually
Redeem Them” WSJ 2/10/98 Expected value
2/11/98 “Surprise! A Home Builder (Finally) Surveys Buyers” WSJ
5/6/98 “In Battle for TV Ads, Cable Is Now the Enemy” WSJ
6/26/98 “Sense and the Census” WSJ
8/3/98 “Networks to Launch a Rival to Nielson Service WSJ
2/8/99 “Can You Trust the Polls? Well, Sometimes.” WSJ
4/13/99
“Is a Web Political Poll reliable? Yes? No? Maybe? WSJ 2/25/00 “Nielson ratings
Spark a Battle Over Just Who Speaks Spanish” WSJ
May 2001 “Do People Mean What They Say? Implications for subjective Survey Data? AER
Papers and Proceedings
10/18/06
“655,00 War Dead? WSJ A20 Cluster sampling
4/17/10
It Is 90% Certain That Unemployment Rose. Or Fell. WSJ A2
5/8/10
In Counting Illegal Immigrants, Certain Assumptions Apply WSJ A2