AP STATISTICS Unit 6: Random Variables
Enduring understanding (Big Idea): Students will understand how to use probability distributions to investigate random phenomena. This includes how to
determine appropriate distributions and find probabilities of occurrence from such distributions, including the binomial and geometric distributions. Students
will also learn the mathematics of random variables, including finding means and standard deviations for both random variables and linear combinations of
independent random variables
Essential Questions:
1. What are the differences, if any, between an unknown variable such as
those used in Algebra and a Random Variable?
2. How do we find the means and variances of discrete and continuous
random variables?
3. If X is a Random Variable, what are the differences, if any, between 2X
and X + X?
4. What are the differences, if any, between the means and variances of the
sum versus the difference of two independent discrete random variables?
5.
6.
7.
8.
What are the differences between the variances of linear combination of two
independent versus not independent discrete random variables?
What is the difference between a combination of any random variables versus a
combination of Normal random variables?
When is it appropriate to use Binomial or Geometric Distributions?
What are formulae for the means and standard deviations of Binomial and
Geometric Distributions?
BY THE END OF THIS UNIT:
Students will know: (See the AP Statistics Course Description page 7 for details)
AP Stats IIIA and B Exploring random phenomena using probability and
simulation
Vocabulary: Binomial Coefficient, Binomial Distribution, Binomial Probability,
Binomial Random Variable, Binomial Setting (BINS), Combining Random Variables,
Continuous Random Variable, Discrete Random Variable, Expected Value, Factorial,
Geometric Distribution, Geometric Probability, Geometric Random Variable,
Geometric Setting (BITS), Independent Random Variables, Linear Transformations
of Random Variables, Mean (Expected Value) of a Binomial Random Variable, Mean
(Expected Value) of a Geometric Random Variable, Mean of a Discrete Random
Variable, Mean of the Difference of Random Variables, Normal Approximation for
Binomial Distributions, Probability Distribution, Random Variable, Sampling
Without Replacement Condition (also known as the 10% condition), Standard
Deviation of a Binomial Random Variable, Standard Deviation of a Discrete Random
Variable, Variance of a Discrete Random Variable, Variance of the Difference of
Random Variables, Variance of the Sum of Independent Random Variables
Unit Resources: (see Fathom, Applets and Glossary in Unit #3)
Learning Tasks: homework - see page 6-6 of the Teacher's Edition
Performance Task: "FRAPPY" (see Formative Assessments below)
Project "TBD"
Students will be able to
1. use probability distributions to investigate possible values of a random
variable
2. calculate and interpret the mean and standard deviation of a discrete
random variable
3. describe the effects of linearly transforming a random variable
4. find the mean and standard deviation of the sum or difference of
independent random variables
5. determine whether to random variables are independent find probabilities
for sums or differences of independent random variables
6. determine whether random variables are binomial or geometric, their
means and standard deviations, and associated probabilities
7. use the normal approximation for a binomial distribution
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
AP STATISTICS Unit 6: Random Variables
CORE CONTENT
Cluster Title: D. Understand and evaluate random processes underlying statistical experiments (S-IC)
S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation
Cluster Title: H. Calculate expected values and use them to solve problems (S-MD)
S.MD.1 (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the
corresponding probability distribution using the same graphical displays as for data distributions.
S.MD.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
S.MD.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated;
find the expected value.
S.MD.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find
the expected value.
Cluster Title: I. Use probability to evaluate outcomes of decisions (S-MD)
S.MD.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance.
b. Evaluate and compare strategies on the basis of expected values.
NCSCOC 3.05 Apply concepts of probability to solve problems http://www.dpi.state.nc.us/curriculum/mathematics/scos/2003/9-12/72apstatistics
AP Stats IIIA4 Discrete random variables and their probability distributions…
AP Stats IIIA6 Mean (expected value) and standard deviation of a random variable
Concepts and Skills to Master (on paper and with technology)
Section 6.1
[Discrete and Continuous Random Variables]
 Discrete random variables
 Standard deviation (and variance) of a discrete random variable
 Mean (expected value) of a discrete random variable
 Continuous Random Variable
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Prior material in this course contained in Units 1 – 5 (i.e. Chapters 1-5)
Academic Vocabulary
Continuous Random Variable, Discrete Random Variable, Expected Value, Mean of a Discrete Random Variable, Probability Distribution, Random Variable,
Standard Deviation of a Discrete Random Variable, Variance of a Discrete Random Variable
Suggested Instructional Strategies


Rely on the Teachers' Edition and the Platinum Resource Binder
The text is excellent and students should have their book (and their calculator) at each
class. The challenge is how to inspire kids to read and study it. There should be class
discussions during each class using material from the book and using the calculator.
Consider assigning the next class' material as a prereading. Especially focus on the boxes
labeled "AP Exam Common Error" in the margins (and also in Appendix A)
Resources
 Textbook Correlation:
Chapter 6 , Introduction and Section 1 (pages
340 - 357)
 Most of the PowerPoint slides on the Instructor's
CD are very helpful.
 Use resources at
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
AP STATISTICS Unit 6: Random Variables






Begin with a discussion of the Case Study on page 339 and lay a foundation for how to
consider the probabilities of chance outcomes (i.e. random guessing by the practitioners)
Do Activities on pages 339 and 340, followed by a technology simulation of the activity.
With all students on their calculators and you on your projection calculator do Technology
Corner 6.1 on page 348 (or other Technology)
Have a Class Discussion of Section 6.2's "Summary" on page 352
The College Board has posted tremendously helpful material
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2151.html
in particular use:
http://apcentral.collegeboard.com/apc/members/exam/exam_information/8357.html
Useful directories for the released questions are at
http://www.apstatsmonkey.com/StatsMonkey/AP_Review.html
for FR look for AP Index 11_12.xls
for MC look for MC_Index.xls - Daren Starnes
Sample Formative Assessment Tasks
Skill-based tasks



CHECK YOUR UNDERSTANDING
(textbook pages 344, 349,)
Use assessments at http://www.stats4stem.org/
Use Released Multiple Choice questions (see the
last Suggested Instructional Strategy above for a
link to the list of MC questions). The column "%
correct” shows the global performance on the AP
Exam for that question. Add a column to the
spreadsheet to keep a record of the questions you
used and when used.
http://apstatsmonkey.com/StatsMonkey/Statsm
onkey.html
 Monitor the AP Statistics List Serve [see
Electronic Discussion Groups on
http://apcentral.collegeboard.com/apc/public/co
urses/teachers_corner/2151.html ] soon to be
an AP community see
https://epl.collegeboard.org/epl/login.do
Problem Tasks

At the End of Section 6.1 do FRAPPY #7.1 [which uses 2001 #2]:
http://www.apstatsmonkey.com/StatsMonkey/FRAPPYs_files/Frappy7.1.pdf (Or as an
alternate, see the last Suggested Instructional Strategy above for a list of free response
questions that you could alternately use as your FRAPPY. Go to
http://www.apstatsmonkey.com/StatsMonkey/AP_Review.html for the FR questions.
Look for the link AP Index 11_12.xls. Add a column to the spreadsheet and keep a record
of the questions you used and when used.)

To Use Frappies: On the first page at
http://apstatsmonkey.com/StatsMonkey/FRAPPYs.html click on the FRAPPY Presentation
to learn about them and how to use them as Formative Assessments. (Click on the Frappy
Presentation to Advance to a new slide. You may also download handout notes using the
link to the right)
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
AP STATISTICS Unit 6: Random Variables
CORE CONTENT
Note this AP topic goes beyond the common core standards
NCSCOC 3.05 Apply concepts of probability to solve problems
http://www.dpi.state.nc.us/curriculum/mathematics/scos/2003/9-12/72apstatistics
AP Stats IIIA6 … Linear transformation of a random variable
AP Stats IIIB Combining independent random variables:
1 Notion of independence versus dependence
2 Mean and standard deviation for sums and differences of independent random variables
Concepts and Skills to Master (on paper and with technology)
Section 6.2
[Transforming and Combining Random Variables]
 Linear transformations
 Combining normal random variables
 Combining random variables
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Prior material in this course contained in Units 1 – 6 (i.e. Chapters 1- Section 6.1)
Academic Vocabulary
Continuous Random Variable, Discrete Random Variable, Expected Value, Mean of a Discrete Random Variable, Probability Distribution, Random Variable,
Standard Deviation of a Discrete Random Variable, Variance of a Discrete Random Variable
Suggested Instructional Strategies






Rely on the Teachers' Edition and the Platinum Resource Binder
The text is excellent and students should have their book (and their calculator) at each
class. The challenge is how to inspire kids to read and study it. There should be class
discussions during each class using material from the book and using the calculator.
Consider assigning the next class' material as a prereading. Especially focus on the boxes
labeled "AP Exam Common Error" in the margins (and also in Appendix A)
With all students on their calculators and you on your projection calculator do
Technology Corner 6.2 on page 376 (or other Technology)
Have a Class Discussion of Section 6.2's "Summary" on page 377
The College Board has posted tremendously helpful material
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2151.html
in particular use:
http://apcentral.collegeboard.com/apc/members/exam/exam_information/8357.html
Useful directories for the released questions are at
http://www.apstatsmonkey.com/StatsMonkey/AP_Review.html
for FR look for AP Index 11_12.xls
for MC look for MC_Index.xls - Daren Starnes
Resources
 Textbook Correlation:
Chapter 6 , Section 2 (pages 358 - 382)
 Most of the PowerPoint slides on the Instructor's
CD are very helpful.
 Use resources at
http://apstatsmonkey.com/StatsMonkey/Statsmo
nkey.html
 Monitor the AP Statistics List Serve [see Electronic
Discussion Groups on
http://apcentral.collegeboard.com/apc/public/co
urses/teachers_corner/2151.html ] soon to be an
AP community see
https://epl.collegeboard.org/epl/login.do
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
AP STATISTICS Unit 6: Random Variables
Sample Formative Assessment Tasks
Skill-based tasks



Problem Tasks

CHECK YOUR UNDERSTANDING (textbook
pages 362, 370 and 372)
Use assessments at http://www.stats4stem.org/
Use Released Multiple Choice questions (see the last
Suggested Instructional Strategy above for a link to the
list of MC questions). The column "% correct”
shows the global performance on the AP Exam for that
question. Add a column to the spreadsheet to keep a
record of the questions you used and when used.

At the End of Section 7.2 do FRAPPY #7.2 [which uses 2002 #3]:
http://apstatsmonkey.com/StatsMonkey/FRAPPYs_files/Frappy7.2.pdf Or as an
alternate, see the last Suggested Instructional Strategy above for a list of free response
questions that you could alternately use as your FRAPPY. Go to
http://www.apstatsmonkey.com/StatsMonkey/AP_Review.html for the FR questions.
Look for the link AP Index 11_12.xls. Add a column to the spreadsheet and keep a
record of the questions you used and when used.)
To Use Frappies: On the first page at
http://apstatsmonkey.com/StatsMonkey/FRAPPYs.html click on the FRAPPY
Presentation to learn about them and how to use them as Formative Assessments. (Click
on the Frappy Presentation to Advance to a new slide. You may also download handout
notes using the link to the right)
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
AP STATISTICS Unit 6: Random Variables
CORE CONTENT
Cluster Title: D. Understand and evaluate random processes underlying statistical experiments (S-IC)
S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation
NCSCOC 3.05 Apply concepts of probability to solve problems
http://www.dpi.state.nc.us/curriculum/mathematics/scos/2003/9-12/72apstatistics
AP Stats IIIA4 … including binomial and geometric
Concepts and Skills to Master (on paper and with technology)
Section 6.3
[Binomial and Geometric Random Variables]
 Binomial - settings, random variables, and probabilities
 Binomial distributions in statistical sampling
 Mean and standard deviation of a binomial distribution
 Geometric - settings, random variables, and probabilities
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Previous Skills and concepts in this course from Chapter 1 through Section 6.2
Academic Vocabulary
Binomial Coefficient, Binomial Distribution, Binomial Probability, Binomial Random Variable, Binomial Setting (BINS), Factorial, Geometric Distribution,
Geometric Probability, Geometric Random Variable, Geometric Setting (BITS), Mean (Expected Value) of a Binomial Random Variable, Mean (Expected
Value) of a Geometric Random Variable, Normal Approximation for Binomial Distributions, Sampling Without Replacement Condition (also known as the
10% condition), Standard Deviation of a Binomial Random Variable
Suggested Instructional Strategies






Resources
Rely on the Teachers' Edition and the Platinum Resource Binder
 Textbook Correlation:
The text is excellent and students should have their book (and their calculator) at each 
Chapter 6, Section 3 (pages. 382 - 411)
class. The challenge is how to inspire kids to read and study it. There should be class
 Most of the PowerPoint slides on the Instructor's
discussions during each class using material from the book and using the calculator.
CD are very helpful.
Consider assigning the next class' material as a prereading. Especially focus on the boxes
 Use resources at
labeled "AP Exam Common Error" in the margins (and also in Appendix A)
http://apstatsmonkey.com/StatsMonkey/Statsmonkey.
With all students on their calculators and you on your projection calculator do
html
Technology Corners 6.3 on pages 388, 389, and 400 (or other Technology)
 Monitor the AP Statistics List Serve [see Electronic
Have Class Discussions of Section 6.3's "Summary" on page 402
Discussion Groups on
The College Board has posted tremendously helpful material (continued)
http://apcentral.collegeboard.com/apc/public/course
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2151.html
in particular use:
s/teachers_corner/2151.html ] soon to be an AP
http://apcentral.collegeboard.com/apc/members/exam/exam_information/8357.html
community see
Useful directories for the released questions are at
https://epl.collegeboard.org/epl/login.do
http://www.apstatsmonkey.com/StatsMonkey/AP_Review.html
for FR look for AP Index 11_12.xls
for MC look for MC_Index.xls - Daren Starnes
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
AP STATISTICS Unit 6: Random Variables
Sample Formative Assessment Tasks
Skill-based tasks






CHECK YOUR UNDERSTANDING (pages 385, 390, 393, and
401)
Use assessments at http://www.stats4stem.org/
Use Released Multiple Choice questions (see the last Suggested
Instructional Strategy above for a link to the list of MC questions) See
suggestion for usage in Unit # 3
Use Quizzes at http://bcs.whfreeman.com/tps4e/#628644__630756__
Do the Chapter 6 AP Statistics Practice Test on page 409
Use Released Multiple Choice questions (see the last Suggested
Instructional Strategy above for a link to the list of MC questions).
The column "% correct " shows the global performance on the AP
Exam for that question. Add a column to the spreadsheet to keep a
record of the questions you used and when used.
Problem Tasks

At the End of Unit 6 do the FRAPPY using 2010B #3. It is
located at:
http://www.apstatsmonkey.com/StatsMonkey/FRAPPYs.html
look on the lower part of the right column for
”Binomial/Geometric RV Binomial, Geometric.zip "
(Or as an alternate, see the other FRAPPYs contained in the ZIP
file.)

To Use Frappies: On the first page at
http://apstatsmonkey.com/StatsMonkey/FRAPPYs.html click on the
FRAPPY Presentation to learn about them and how to use them as
Formative Assessments. (Click on the Frappy Presentation to
Advance to a new slide. You may also download handout notes
using the link to the right)
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.