Week 1:  Mon, Aug 24 
Wed, Aug 26 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Aug 28 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 2:  Mon, Aug 31 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Sep 2 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Sep 4 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 3:  Mon, Sep 7 Labor Day (no class) 
Wed, Sep 9 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Sep 11 Probability Mass Functions and CDFs probability mass function (pdf / video) babies example (pdf / video) CDF (pdf / video) CDF is nondecreasing (pdf / video) CDF is nondecreasing (v 2) (pdf / video) "first success" example (pdf / video) cookie example (pdf / video) minimum / maximum example (pdf / video) Practice Problems and Practice Solutions 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 4:  Mon, Sep 14 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Sep 16 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Sep 18 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 5:  Mon, Sep 21 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Sep 23 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 midSeptember Review and midSeptember Review Solutions 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Sep 25 Geometric Random Variables Geometric (pdf / video) lefthanded 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 6:  Mon, Sep 28 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) lefthanded example (pdf / video) sums of independent Negative Binomials (pdf / video) Practice Problems and Practice Solutions 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Sep 30 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Oct 2 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 7:  Mon, Oct 5 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Oct 7 Review for Midterm Exam 1 
Fri, Oct 9 Midterm Exam 1 

Week 8:  Mon, Oct 12 October Break (no class) 
Wed, Oct 14 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Oct 16 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 9:  Mon, Oct 19 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Oct 21 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions Dr. George McCabe (Associate Dean in College of Science) will visit our class 
Fri, Oct 23 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 10:  Mon, Oct 26 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Oct 28 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Oct 30 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 11:  Mon, Nov 2 Second day of study of exponential random variables (same notes as those from Friday, Oct 30) 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Nov 4 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Nov 6 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.) 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 

Week 12:  Mon, Nov 9 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Wed, Nov 11 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions 
Fri, Nov 13 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 2014 InClass Problem Set 1 and InClass Problem Set Solutions 1 2014 InClass Problem Set 2 and InClass Problem Set Solutions 2 2015 InClass Problem Set and InClass Problem Set Solutions  
Week 13:  Mon, Nov 16 Review for Midterm Exam 2 
Wed, Nov 18 Midterm Exam 2 
Fri, Nov 20 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions  
Week 14:  Mon, Nov 23 More practice with variance, covariance, and correlation (please see practice problems from Fri, Nov 20; it is worthwhile to practice these topics for 2 days) 2015 InClass Problem Set and InClass Problem Set Solutions  Wed, Nov 25 Thanksgiving Vacation (no class) 
Fri, Nov 27 Thanksgiving Vacation (no class) 

Week 15:  Mon, Nov 30 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions  Wed, Dec 2 Markov and Chebyshev Inequalities Markov inequality (pdf / video) Examples (pdf / video) Chebyshev's inequality (pdf / video) Examples (pdf / video) Practice Problems and Practice Solutions 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions  Fri, Dec 4 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions  
Week 16:  Mon, Dec 7 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.) 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions  Wed, Dec 9 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 2014 InClass Problem Set and InClass Problem Set Solutions 2015 InClass Problem Set and InClass Problem Set Solutions  Fri, Dec 11 Review for Final Exam 

Final exam date/time/location: Mon 12/14, during 7:00p  9:00p, in FRNY G140 