MAT 330/681: Probability

Lehman College, City University of New York

Fall 2017

This textbook has been used for this course for multiple years, so it should be possible to find a used copy.

Other free online textbooks on Probability:

In-class Quizzes 25%

Midterm 25%

Final Exam 35%

You must take and pass the final to pass the course. The lowest three quizzes and lowest three problem set grades will be dropped.

Date: |
Topics: |
Reading: |
Homework assigned: |
Due: |

#1
Tues 29 August |
Review of syllabus and academic integrity code. Introduction, permutations | Syllabus; Textbook sections 1.1-1.3; | HW 1a: Chapter 1 #1,6,8,9 (from Problems section) | |

#2
Thurs 31 August |
Combinations | Textbook section 1.4 Permutations vs Combinations How to Understand Combinations Using Multiplication Why do we multiply combinations? Visualizing permutations and combinations |
HW 1b: Chapter 1 #12,20,24 (from Problems section); #7 (from Theoretical Exercises) | |

Mon 4 September | CUNY: No classes (Labor Day) | |||

#3
Tues 5 September |
Multinomial coefficients and balls in urns | Textbook sections 1.5-1.6 | HW 2a: Chapter 1 #25,27,31 (from Problems section) | |

#4
Thurs 7 September |
Sample spaces and events | Textbook sections 2.1, 2.2, 2.5 Visualizing set relationships |
HW 2b: Chapter 2 #1,3,19 (from problems section) ; How many different ways are there to walk from Columbus Circle (at 59th St and 8th Ave.) to Grand Central Station (at 46nd St. and Park Ave.)? Assume you can only walk down Streets and Avenues. That is, assume you cannot walk along Broadway, through alleys, or cut through buildings. You may need to look at a map. | Homework 1a and 1b due at beginning of class |

#5
Tues 12 September |
Basic probability propositions | Textbook sections 2.3, 2.4 | HW 3a: Chapter 2 # 10, 13 (from problems section); #6, 13 (from Theoretical Exercises) | Quiz on Sections 1.1-1.4 (HW 1a and 1b) |

#6
Thurs 14 September |
Basic probability propositions cont'd | Textbook sections 2.3, 2.4 | HW 3b: Chapter 2 # 8, 15,19 (from problems section); #7 (from Theoretical Exercises) | Homework 2a and 2b due at beginning of class |

Tues 19 September | CUNY: Classes follow a Thurs schedule | |||

#7
Tues 19 September |
Conditional Probability | Textbook section 3.2 Conditional probability explained visually Another visualization of conditional probability |
HW 3c: Chapter 3 # 1, 2, 4, 17 (from problems section) | |

20-22 September | CUNY: No classes | |||

#8
Tues 26 September |
Bayes's Formula and Independent events | Textbook sections 3.3 and 3.4 Explanation and intuition of Bayes' Formula |
HW 4a: Chapter 3 #19, 35, 59 (from problems section); 2 (from theoretical exercises) | Quiz on Sections 1.5, 1.6, 2.1, 2.2, 2.5 (HW 2a and 2b) |

29-30 September | CUNY: No classes | |||

#9
Thurs 28 September |
Independent Events Cont'd, Introduction to Random Variables | Textbook sections 3.4, 4.1 | HW 4b: Chapter 3 #33, 55, 62 (from problems section); 9 (from theoretical exercises) | Homework 3a, 3b, and 3c due at beginning of class |

#10
Tues 3 October |
Introduction to Random Variables | Textbook section 4.1 Visualizing random variables |
HW 5a: Chapter 4 #1, 5, 6, 7 (from problems section) | Quiz on Sections 2.3, 2.4, and 3.2 (HW 3a, 3b, and 3c) |

#11
Thurs 5 October |
Discrete Random Variables, Expectation of a Random Variable | Textbook sections 4.2, 4.3, 4.4 Visualizing expectation |
HW 5b: Chapter 4 #18, 19, 21, 25 (from problems section) | Homework 4a and 4b due at beginning of class |

9 October | CUNY: No classes (Columbus Day) | |||

#12
Tues 10 October |
Variance of random variables | Textbook sections 4.4, 4.5 Visualizing variance |
HW 6a: Chapter 4 #28, 35, 37 (from problems section); #8 (from theoretical exercises) | Quiz on Sections 3.3 and 3.4 (HW 4a and 4b) |

#13
Thurs 12 October |
Variance Continued, Bernoulli and Binomial Random Variables Introduced | Textbook sections 4.5 and 4.6 | HW 6b: Question 1: Chapter 4 #38 (from problems section) Questions 2 and 3: Let X and Y be discrete random variables with the same possible values. Prove the following using the definition from class of expected value of a random variable or a function of a random variable: (i) E[X + Y] = E[X] + E[Y] (ii) E[aX^2] = aE[X^2] Question 4: Let B be a Bernoulli random variable with p = 0.4. What is E[B] and Var(B)? |
Homework 5a and 5b due at beginning of class |

#14
Tues 17 October |
Binomial Random Variables continued | Textbook sections 4.6 Visualization of binomial distribution |
HW 7a: Chapter 4 #40, 41, 59 (submit on or by beginning of class on Thurs. Oct. 26 for extra credit) Note: #44 has been removed | Quiz on Sections 4.1, 4.2, 4.3, 4.4 (HW 5a and 5b) |

#15
Thurs 19 October |
Review for midterm | Homework 6a and 6b due at beginning of class | ||

#16
Tues 24 October |
Midterm Exam | |||

#17
Thurs 26 October |
Binomial random variables continued, Poisson random variables | Textbook sections 4.6, 4.7 Visualizing Poisson and other distributions |
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#18
Tues 31 October |
Went over midterm | HW 8a: Chapter 4 #48, 51, 55, 58 (from problems section) | No quiz | |

#19
Thurs 2 November |
Poisson randome variables continued, introduction to continuous random variables | Textbook sections 4.7, 5.1 |
HW 8b: Chapter 5 #1, 3, 4, 13 (from problems section) Note: The cumulative distribution function for continuous random variables is defined the same way as for discrete random variables: F(a) = P{X ≤ a} |
Homework 7a and 7b due at beginning of class |

#20
Tues 7 November |
Expectation and Variance of Continuous Random Variables, Uniform Random Variable | Textbook sections 5.1, 5.2, 5.3 | HW 9a: Chapter 5 #2, 5, 6, 7 (from problems section) | Quiz on Sections 4.6 (HW 7a |

#21
Thurs 9 November |
Uniform Random Variable, Normal Random Variable, |
Textbook sections 5.3, 5.4, Normal distribution interactive graph Normal distribution calculator |
HW 9b: Chapter 5 #10, 14 (Prop. 2.1 is on page 191), |
Homework 8a and 8b due at beginning of class |

10 November | Last day to withdraw from class with a grade of W | |||

#22
Tues 14 November |
Normal random variable continued, Jointly distributed random variables | Textbook section 5.4 | HW 10a: Chapter 5 #15, 16, 19, 21 (from problems section) | Quiz on Sections 4.7, 5.1 (HW 8a and 8b) |

#23
Thurs 16 November |
Jointly distributed random variables | Textbook section 6.1 | HW 10b: Chapter 6 # 1, 2a, 3a, 4 (from problems section) | Homework 9a and 9b due at beginning of class |

21 November | Classes follow Friday schedule | |||

23-25 November | Thanksgiving Recess: College Closed | |||

#24
Tues 28 November |
Jointly distributed random variables, independent random variables | Textbook sections 6.1 and 6.2 | HW 11a: Chapter 6 #6, 9, 23 (a, b, and c only; hint: to find E[X], find the marginal distribution of X), 40b (from problems section) | Quiz on Sections 5.1, 5.2, 5.3 (HW 9a and 9b) |

#25
Thurs 30 November |
Expectation of Sums of Random Variables, Chebyshev's Inequality and the Weak Law of Large Numbers | Textbook section 7.2, 8.2 Visualization of Weak Law of Large Numbers (2nd figure) |
HW 11b: Chapter 8 #1, 2 (from problems section); Let X and Y be independent random variables. Prove that E[XY] = E[X]E[Y] when a) X and Y are discrete random variables with joint pmf p(x,y), and b) X and Y are continuous random variables with joint pdf f(x,y). Note that this is not true when X and are Y are dependent. | Homework 10a and 10b due at beginning of class |

#26
Tues 5 December |
Central Limit Theorem (CLT) | Textbook section 8.3 Visualization of CLT for Bernoulli r.v. Visualization of CLT from statistics perspective Another visualization of CLT |
HW 12a: Chapter 8 #6, 13ab, 15, 16 (from problem section) (I recommend doing #6 last) Due: Tues. Dec. 12 |
Quiz on Sections 5.4 and 6.1 (HW 10a and 10b) |

#27
Thurs 7 December |
Markov chains (not on final exam) and review for final exam | Markov Chains explained visually | Homework 11a and 11b due at beginning of class | |

#28
Tues 12 December |
Review for final exam | Homework 12a due at beginning of class | ||

Thurs 14 December | Final exam 1:30pm - 3:30pm |