STATS/MATH 425 SECTION 007

Introduction to Probability

Fall 2014

Class Information

Instructor Information

Name: Ambuj Tewari

Office: 454 West Hall

Office Hours: Tuesdays and Thursdays, 9 am -- 10 am

Email: tewaria@umich.edu

GSI information

Name: Yun-Jhong Wu

Office Hours & Location: Wednesdays, 7 pm -- 8:30 pm in SLC (1720 Chemistry)

Email: yjwu@umich.edu

Grading

The final grade in the course will be determined by your scores in homeworks, one midterm exam, and one final exam using the weights given below.

Accommodations for Students with Disabilities

If you think you need an accommodation for a disability, please let me know at your earliest convenience. Some aspects of this course, the assignments, the in-class activities, and the way the course is usually taught may be modified to facilitate your participation and progress. As soon as you make me aware of your needs, we can work with the Office of Services for Students with Disabilities (SSD) to help us determine appropriate academic accommodations. SSD (734-763-3000; http://www.umich.edu/sswd) typically recommends accommodations through a Verified Individualized Services and Accommodations (VISA) form. Any information you provide is private and confidential and will be treated as such.

Academic Integrity

Please familiarize yourself with the LSA Community Standards of Academic Integrity. The College of LSA expects all of its members to uphold these Standards.

Schedule

The tentative schedule for the semester is given below. It will most likely change as we move along. References of the form (x.y) refer to sections in the textbook.

Day

Plan

Sep 2

  • Basic principle of counting (1.2)
  • Examples 2a, 2c, 2e
  • Permutations (1.3)
  • Examples 3b, 3d

Sep 4

  • Combinations (1.4)
  • Basic formula
  • Identity (4.1)
  • Example 4b
  • Combinatorial proof of Binomial theorem
  • Example 4e
  • HW 1 out

Sep 9

  • Multinomial Coefficients (1.5)
  • Examples 5a, 5b, 5c

Sep 11

  • Sample space and events (2.2)
  • Examples of experiments and their sample spaces
  • Operations (unions, intersections, complements) on events
  • de Morgan’s laws
  • HW 1 due
  • HW 2 out

Sep 16

  • Axioms of probability (2.3)
  • Examples 3a, 3b
  • Proposition 4.1, 4.2 and 4.3
  • Example 4a
  • Inclusion-Exclusion principle (2.4)

Sep 18

  • Sample spaces having equally likely outcomes (2.5)
  • Examples 5a, 5b, 5c
  • HW 2 due (deadline extended to Sep 23)
  • HW 3 out

Sep 23

  • Sample spaces having equally likely outcomes (2.5) continued
  • Example 5d
  • Conditional Probabilities (3.2)
  • Definition (Eq. (2.1))
  • Examples 2b, 2d

Sep 25

  • Conditional Probabilities (3.2) continued
  • The multiplication rule
  • Example 2g
  • Bayes’ Formula (3.3)
  • Examples 3a (both parts), 3c, 3d
  • HW 3 due
  • HW 4 out

Sep 30

  • Bayes’ Formula (3.3) continued
  • Eq. (3.4)
  • Proposition 3.1
  • Examples 3k
  • Independent Events (3.4)
  • Example 4b

Oct 2

  • Independent Events (3.4) continued
  • Example 4e
  • Independence of multiple events
  • HW 4 due
  • HW 5 out

Oct 7

  • Independent Events (3.4)
  • Example 4g, 4h
  • Random variables (4.1)
  • Example 1a

Oct 9

  • Midterm Review
  • HW 5 due
  • HW 6 out

Oct 14

NO CLASS (Fall Study Break)

Oct 16

MIDTERM EXAM 

10:10-11:30 (80 minutes) in 120 Dennison

Oct 21

  • Random variables (4.1)
  • Example 1d
  • Discrete random variables (4.2)
  • Probability Mass Function
  • Example 2a
  • Cumulative Distribution Function

Oct 23

  • Expected value (4.3)
  • Examples 3a
  • Expected value of a function of a random variable (4.4)
  • Example 4a, Proposition 4.1, Corollary 4.1
  • Definition of moments
  • HW 6 due
  • HW 7 out

Oct 28

  • Variance (4.5)
  • Definition, alternative formula
  • Example 5a
  • Standard deviation
  • Bernoulli random variables (4.6)
  • Eq. (6.1)
  • Expectation and variance of Bernoulli random variables
  • Binomial random variables (4.6)
  • Eq. (6.2)

Oct 30

  • Binomial random variables (4.6) continued
  • Example 6b
  • Expectation and variance (4.6.1)
  • Poisson random variable (4.7)
  • Eq. (7.1)
  • Example 7a
  • MID-SEMESTER FEEDBACK
  • HW 7 due
  • HW 8 out

Nov 4

  • Poisson random variable (4.7) continued
  • Poisson as an approximation to Binomial when n is large, p small
  • Example 7b
  • Expected value of sums (4.9)
  • Corollary 9.2
  • Example 9c

Nov 6

  • Introduction to continuous random variables (5.1)
  • Relationship between pdf and cdf
  • The two equations right after Example 1c in Section 5.1
  • Expectation and variance (5.2)
  • Proposition 2.1
  • Corollary 2.1
  • HW 8 due
  • HW 9 out

Nov 11

  • The uniform random variable (5.3)
  • Example 3a
  • Normal random variables (5.4)
  • “an important fact” near the middle of page 188
  • Example 4a

Nov 13

  • Exponential random variables (5.5)
  • Example 5a
  • Memoryless property
  • Hazard Rate Functions (5.5.1)
  • Definition & interpretation
  • Computing cdf given hazard rate function: Eq (5.4)
  • HW 9 due
  • HW 10 out

Nov 18

  • NO CLASS

Nov 20

  • Joint distribution functions (6.1)
  • joint cdf
  • joint pmf (for jointly discrete random variables)
  • joint pdf (for jointly continuous random variables)
  • Examples 1a, 1c, 1e
  • HW 10 due

Nov 25

  • Recap of discrete RVs, continuous RVs, jointly discrete RVs and jointly continuous RVs
  • Independent random variables (6.2)
  • Example 2c
  • HW 11 out

Nov 27

  • NO CLASS (Thanksgiving break)

Dec 2

  • Independent random variables (6.2) continued
  • Proposition 2.1
  • Example 2f
  • Normal random variables (5.4) continued
  • cdf of a standard normal
  • Example 4b

Dec 4

  • Normal approximation to the Binomial (5.4.1)
  • Example 4g
  • Conditional distributions: Discrete case (6.4)
  • Example 4a
  • Conditional distributions: Continuous case (6.5)
  • Examples 5a
  • HW 11 due

Dec 9

  • Expectation of sums (7.2)
  • Proposition 2.1
  • Example 2a
  • Equation (2.1)
  • Review

Dec 16

FINAL EXAM (1:30-3:30 in 120 Dennison)