Probability
STAT/MA 41600, Fall 2014

Class times: MWF, 9:30 AM -- 10:20 AM, in UNIV 003
(STAT 41600-002; Banner CRN 29227)
(MA 41600-151; Banner CRN 45603)

Professor: Mark Daniel Ward
Email: mdw@purdue.edu
Office: MATH 540
Phone: 765-496-9563

Office hours: Dr. Ward is always happy to meet with students.
Dr. Ward is available for walk-in or scheduled appointments anytime, throughout the week.
He is also always guaranteed to be available MWF, 7:30 AM -- 8:20 AM, in MATH 540.

Grader: Botao Hao
Email: hao22@purdue.edu

Course description: click here

Course policy: click here

Midterm exam dates: Friday, October 10; Friday, November 21

Final exam date/time/location: Thursday, December 18, from 8 AM to 10 AM in BRNG 2290.

Plan to be present at all exams. Plan to be present for class every day.

Homework: All assigned homework problems are posted in the calendar grid below. Homework solutions will be collected in class on the due date. Homework solutions will be posted by the instructor in the calendar grid too. Interesting links? Dr. Ward is happy to post them here:
Outline of Topics
Week 1: Mon, Aug 25
Wed, Aug 27
Outcomes, Events, and Sample Spaces
definitions (pdf / video)
dice example (pdf / video)
coin flip example (pdf / video)
DeMorgan's First Law (pdf / video)
DeMorgan's Second Law (pdf / video)
how many events (pdf / video)
sample spaces (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Aug 29
Probability
probability rules (pdf / video)
probability of empty set (pdf / video)
union of finitely many events (pdf / video)
equally likely outcomes (pdf / video)
more equally likely outcomes (pdf / video)
examples (pdf / video)
setminus and partition (pdf / video)
inclusion/exclusion (pdf1 / pdf2 / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 2: Mon, Sep 1
Labor Day (no class)
Wed, Sep 3
Independent Events
independent events (pdf / video)
independence / disjointness (pdf1 / pdf2 / video)
independence of collections (pdf / video)
examples / good vs bad trials (pdf / video)
something good before something bad (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Sep 5
Conditional Probability
conditional probability (pdf / video)
independence (pdf / video)
dice example (pdf / video)
coin example (pdf / video)
card example (pdf / video)
distributive laws (pdf / video)
conditional probability laws (pdf / video)
symmetry examples (pdf / video)
another coin example (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 3: Mon, Sep 8
Bayes' Theorem
Bayes' Theorem (pdf / video)
example (pdf / video)
Bayes' Theorem (v 2) (pdf / video)
example (pdf / video)
Bayes' Theorem (v 3) (pdf / video)
example (pdf / video)
Bayes' Theorem (v 4) (pdf / video)
probability of an intersection (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Sep 10
Random Variables;
Discrete Versus Continuous
random variable (pdf / video)
dice example (pdf / video)
babies example (pdf / video)
discrete versus continuous (pdf / video)
probabilities (pdf / video)
indicators (pdf / video)
coin example (pdf / video)
outcomes with probability 0 (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Sep 12
Probability Mass Functions
and CDFs
probability mass function (pdf / video)
babies example (pdf / video)
CDF (pdf / video)
CDF is non-decreasing (pdf / video)
CDF is non-decreasing (v 2) (pdf / video)
"first success" example (pdf / video)
cookie example (pdf / video)
minimum / maximum example (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 4: Mon, Sep 15
Joint Distributions;
Independence and Conditioning
joint mass (pdf / video)
baby example (pdf / video)
finding single variable mass (pdf / video)
conditional mass (pdf / video)
independent random variables (pdf / video)
dice example (pdf / video)
indicators (pdf / video)
joint mass of a collection (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Sep 17
Expected Values of
Discrete Random Variables
expected value (pdf / video)
using outcomes (pdf / video)
weighted sum (pdf / video)
hat example (pdf / video)
trials until first success (pdf / video)
calculus review (pdf / video)
maximum of two dice (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Sep 19
Expected Values of
Sums of Random Variables
expected value of sum (pdf / video)
baby example revised (pdf / video)
hat example revised (pdf / video)
trials until first success revised (pdf / video)
discrete expected value, alternative (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 5: Mon, Sep 22
Expected Values of
Functions of Random Variables;
Variance
exp value of function of r.v. (pdf / video)
baby example (pdf / video)
variance (pdf / video)
baby example (pdf / video)
nice variance fact (pdf / video)
more nice variance facts (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Sep 24
Bernoulli Random Variables
Binomial Random Variables
Bernoulli (a.k.a. indicator) (pdf / video)
mass and CDF (pdf / video)
non 0/1 application (pdf / video)
Binomial (pdf / video)
mass (pdf / video)
expected value; variance (pdf / video)
baby example (pdf / video)
card example (pdf / video)
sums of independent Binomials (pdf / video)
Practice Problems and Practice Solutions
More Practice Problems and More Practice Solutions
mid-September Review and mid-September Review Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Sep 26
Geometric Random Variables
Geometric (pdf / video)
left-handed example (pdf / video)
expected value; variance (pdf / video)
number of failures (pdf / video)
inequalities (pdf / video)
memoryless property (pdf / video)
what kind of random variable? (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 6: Mon, Sep 29
Negative Binomial Random Variables
Negative Binomial (pdf / video)
mass example: 4th success (pdf / video)
two card and dice examples (pdf / video)
expected value; variance (pdf / video)
left-handed example (pdf / video)
sums of independent Negative Binomials (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Oct 1
Poisson Random Variables
Poisson (pdf / video)
automobile examples (pdf / video)
inequalities (pdf / video)
expected value; variance (pdf / video)
(one line from the video above is corrected in the notes)
sums of independent Poissons (pdf / video)
approximations to Binomials (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Oct 3
Hypergeometric Random Variables
Hypergeometric (pdf / video)
CD example (pdf / video)
expected value of the square (pdf / video)
CD example (continued) (pdf / video)
variance (pdf / video)
Binomial approximation (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 7: Mon, Oct 6
Discrete Uniform Random Variables; and Counting
Discrete Uniform (pdf / video)
Counting; equally likely outcomes (pdf / video)
multiplying probabilities (pdf / video)
card example (pdf / video)
another card example (pdf / video)
pick 10 items from 4 types (pdf / video)
seating arrangements (pdf / video)
Practice Problems and Practice Solutions
More Practice Problems and More Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Oct 8
Review for Midterm Exam 1
Fri, Oct 10
Midterm Exam 1
Week 8: Mon, Oct 13
October Break (no class)
Wed, Oct 15
Continuous Random Variables
probability density functions (pdf / video)
example with exponential decrease (pdf / video)
cumulative distribution functions (pdf / video)
relationship between density and CDF (pdf / video)
CDF example (pdf / video)
another CDF example (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Oct 17
Jointly Distributed Continuous Random Variables
joint density and joint CDF (pdf / video)
example with exponential decrease (pdf / video)
example continued (pdf / video)
constant joint density (pdf / video)
density from joint density (pdf / video)
another example (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 9: Mon, Oct 20
Independent Continuous Random Variables
definitions (pdf / video)
example (pdf / video)
caveat: domains from independence (pdf / video)
example: minimums (pdf / video)
example with dependence (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Oct 22
Conditional Distributions for Continuous Random Variables
conditional probability density functions (pdf / video)
example with a conditional density (pdf / video)
example: finding a conditional density (pdf / video)
second example (pdf / video)
another example (pdf / video)
example continued (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Oct 24
Expected Values of Continuous Random Variables
definition of expected value (pdf / video)
example (pdf / video)
sanity check, and bounds (pdf / video)
exponential example (pdf / video)
uniform example (pdf / video)
another example (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 10: Mon, Oct 27
Expected Values of
Functions of Random Variables;
Variance
definitions (pdf / video)
example with constant density (pdf / video)
example with polynomial density (pdf / video)
expected value of the reciprocal (pdf / video)
linearity (pdf / video)
sums (pdf / video)
products (pdf / video)
facts about the variance (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Oct 29
Continuous Uniform
Random Variables;
density; CDF (pdf / video)
expected value; variance (pdf / video)
example (pdf / video)
conditioning (pdf / video)
linearity (pdf / video)
minimums (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Oct 31
Exponential
Random Variables;
density; CDF (pdf / video)
expected value; variance (pdf / video)
joint probability density function (pdf / video)
memoryless property (pdf / video)
minimums (pdf / video)
more about minimums (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 11: Mon, Nov 3
Second day of study
of exponential random variables
(same notes as those from Friday, Oct 31)
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Nov 5
Gamma Random Variables
definition and comparisons (pdf / video)
visualization (pdf / video)
density; CDF; mean; variance (pdf / video)
example of recognizing density (pdf / video)
calculating probability (pdf / video)
example sum of Exponentials (pdf / video)
more facts about Gammas (pdf / video)
In the 4th video and note above,
the integral has bounds written as 0 and 1,
but the bounds should be 0 and +infinity.
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Nov 7
Beta Random Variables
definition; density (pdf / video)
expected value; variance (pdf / video)
plots of the density (pdf / video)
example: density and CDF (pdf / video)
example: probabilities, mean, variance (pdf / video)
deriving the expected value (pdf / video)
deriving the variance (pdf / video)
conditional probability (pdf / video)
(No practice problems available for this section;
Dr Ward did not give a problem set on
Beta random variables in the past.)
In-Class Problem Set and In-Class Problem Set Solutions
Week 12: Mon, Nov 10
Normal Random Variables
definition; density (pdf / video)
expected value; variance (pdf / video)
linear transformation (pdf / video)
scaling and shifting to standard Normal (pdf / video)
how to use CDF table (pdf / video)
standard deviations (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Nov 12
Sums of Independent
Normal Random Variables
Sums of Indep. Normals are Normal (pdf / video)
scaling and shifting to standard Normal (pdf / video)
CDF example 1 (pdf / video)
CDF example 2 (pdf / video)
threshhold example (pdf / video)
centered interval example (pdf / video)
adding two kinds of indep. Normals (pdf / video)
difference of two Normals (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Nov 14
Central Limit Theorem (part 1)
Laws of Large Numbers (pdf / video)
Central Limit Theorem (pdf / video)
CLT with continuous Uniforms (pdf / video)
CLT with Gamma (pdf / video)
CLT with Binomial (pdf / video)
CLT with Bernoullis (pdf / video)
CLT with Poisson (pdf / video)
In the 4th video and note above,
instead of 620 and 630, I intended to
write 122.75 and 127.75, respectively.
Practice Problems and Practice Solutions
More Practice Problems and More Practice Solutions
Even More Practice Problems and Even More Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 13: Mon, Nov 17
Central Limit Theorem (part 2)
(same notes as those from Friday, Nov 14)
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Nov 19
Review for Midterm Exam 2
Fri, Nov 21
Midterm Exam 2
Week 14: Mon, Nov 24
Variance of Sums;
Covariance; Correlation
Why covariance? (pdf / video)
Variance vs covariance (pdf / video)
Variance of a sum (pdf / video)
More facts about covariance (pdf / video)
Hat problem (pdf / video)
Continuous example (pdf / video)
Covariance is linear (pdf / video)
Correlation (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Nov 26
Thanksgiving Vacation (no class)
Fri, Nov 28
Thanksgiving Vacation (no class)
Week 15: Mon, Dec 1
Conditional Expectation
Conditional expectation (pdf / video)
Dice example (pdf / video)
Exponential example (pdf / video)
Example continued (pdf / video)
Conditional vs independent (pdf / video)
Tuition example (pdf / video)
Poisson splitting (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Dec 3
Markov and Chebyshev Inequalities
Markov inequality (pdf / video)
Examples (pdf / video)
Chebyshev's inequality (pdf / video)
Examples (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Dec 5
Order Statistics
Order Statistics (pdf / video)
Example (pdf / video)
General formula (pdf / video)
Revisit earlier example (pdf / video)
Application to Uniforms (pdf / video)
Density of a specific Order Statistic (pdf / video)
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Week 16: Mon, Dec 8
Moment Generating Functions
Generating functions (pdf / video)
Binomials (pdf / video)
Poissons (pdf / video)
Continuous Uniform (pdf / video)
Extensions (pdf / video)
(No practice problems available for this section;
Dr Ward did not give a problem set on
Moment Generating Functions in the past.)
In-Class Problem Set and In-Class Problem Set Solutions
Wed, Dec 10
Transformations of One
or Two Random Variables
One variable example (pdf / video)
Another example (pdf / video)
Two variable examples to be distributed in class
Practice Problems and Practice Solutions
In-Class Problem Set and In-Class Problem Set Solutions
Fri, Dec 12
Review for Final Exam
Final exam date/time/location: Thursday, December 18, from 8 AM to 10 AM in BRNG 2290.


This material is based upon work supported by the National Science Foundation under Grant Numbers 0939370, 1140489, 1246818. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.