Math 50

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This schedule is subject to changes throughout the course at my discretion. Readings will be finalized the Friday before they are covered in class. The textbooks references are:

Supplementary material: Material in italics is meant to reenforce your understanding of the core course material, but you will not need to know the specific definitions/theorems/examples for the exams.

Regarding my notes: My course notes are meant as a supplement to the textbook readings and other resources. They are not a self-contained reference for the course material and will inevitably contain some mistakes. I recommend using the notes to get an idea of what concepts/examples are emphasized and then reviewing this material in the textbook.

Please see canvas for section specific updates and assignment due dates. This schedule will be updated throughout the term. While I will attempt to stay a few weeks ahead in terms of planning the readings, readings will be finalized roughly one week before the material is covered.

Week 1: Discrete Probability and Monte Carlo Simulations

Topics:

  • Familiarity with basic concepts in probability (events, probability distribution) (Monday)
  • Independence and conditioning (Wednesday)
  • Computation: Basics of Python programming (arrays, Dataframes (moved to week 3), plotting), The concept of Monte Carlo simulation (Friday)

Class material

Reading:

  • ER:
    • 1.1 (Intro)
    • 1.2 (Probability models)
    • 1.3 (Properties of probability models)
    • 2.1 (Random variables): Definition 2.1.1
    • 1.5/2.8 (Conditional probability): Definition 1.5.1, Theorem 1.5.1, Theorem 1.5.2, Definitions 1.5.2 and 1.5.3
  • ISP:
    • 2.3 (python tutorial) -- I use np.random instead of np.random.default_rng.
    Other:
Week 2: iid Sums, Binomial and CLT

Topics:

  • Expectations and variances, conditional expectation (Monday)
  • Binomial distribution, LLN (Monday)
  • Computation: Monte Carlo simulation, histogram, numerical illustration of CLT (Wednesday)
  • Continuous probability distributions and probability density , Central Limit Theorem and Normal distribution (Friday)

Class material

Reading:

  • ER:
    • 3.1 and 3.2 (Expectations)
    • 3.5 (conditional expectation)
    • 3.3 (Variance and covariance)
    • 2.3 (Discrete distributions)
    • 2.4 (Continuous)
    • 4.2.1/4.4.1 (Law of large numbers/Central Limit Theorem): You will not need to know the more technical definitions in the textbook, only the intuition behind these results. The CLT theorem video referenced below is extremely helpful for this.
  • Addition resources (from 3Blue1Brown):

    • Assignments due:

    Week 3: Working with Normal RVs, Least squares LR

    Topics:

    • Properties of Normal random variables (Monday)
    • Single-predictor regression as conditional model (Monday)
    • Correlation coefficients, R-squared, regression to the mean (Wednesday)
    • Least squares (Wednesday)
    • Computation: Simulating regression models and working with tabular data (Dataframes) (Friday)

    Class material

    Reading:

    • ER:
      • 4.6 (Properties of Normal distribution)
      • Definition 3.3.3 covariance
      • 10.1 (related variables): Example 10.1.1
      • 10.3.2 (Simple lineage regression model): Example 10.3.3. Use slightly different notation (e.g. instead of b I write a hat over the regression coefficient to indicate its estimate). You can skip Theorem 10.3.2, 10.3.3 and 10.3.4 for now.

    Assignments due:

    • HW2
    • HW1 Self-evaluation
    Week 4: Other aspects of single predictor LR

    Topics:

    • Computation: Finish regression examples in python, coefficient of determination
    • More on coefficient of determination, estimators, standard error (Wednesday/Friday)
    • Computation: regression with statsmodels, visualizing confidence intervals in regression (Friday)

    Reading:

    • ROS
      • Chapter 4: Read the entire chapter (it's not too technical), but 4.2 and 4.4 are especially important.
    • ER (OPTIONAL): These are optional if you would prefer a more technical treatment.
      • 6.1 and 6.3
    • ISP (OPTIONAL): This is helpful if you would like additional examples in Python.
      • 3.1 (Linear regression)

    Class material

    Assignments due:

    • HW3 Due date pushed to Week 5
    • HW2 Self-evaluation
    Week 5: Hypothesis testing for LR MIDTERM (Oct 16)

    Topics:

    • Midterm review (Monday)
    • Introduction to regression with multiple predictors (Friday)
    • Computation: $p$-values, Performing multivariate regression in statsmodels and data visualization (Friday)

    Reading:

    • No new reading

    Assignments due:

    Week 6: Multiple predictor LR I

    Topics:

    • No class monday
    • Effects of adding predictors to regression models (Wednesday)
    • Interpreting regression coefficients and model building considerations (Wednesday)
    • Computation: Examples in python (Wednesday)

    Reading:

    • ROS:
      • Ch. 10: Ignore the r code and skip 10.5,10.8 and 10.9
    • ISP (OPTIONAL): This is helpful if you would like additional examples in Python beyond my notebooks.
      • 3.1 (Linear regression)

    Class material

    Assignments due:

    • No HW due
    • HW3 Self-evaluation
    Week 7: Multiple predictor LR II

    Topics:

    • Simpsons paradox (Monday)
    • Catagorical predictors/dummy variables (Monday/Wednesday)
    • Interactions (Wednesday)
    • Computation: Hands on examples in statsmodels

    Reading/notes:

    Assignments due:

    Week 8: Model assessment and nonlinear models

    Topics:

    • Bias variance tradeoff, overfitting, double descent (Monday)
    • Cross validation (Monday)
    • Regularization (Wednesday/Friday)
    • Laplace rule of succession

    Reading/notes:

    Additional resources

    Assignments due:

    • HW4 Self-evaluation
    Week 9/10: Fourier models, Bayesian Inference, other topics

    Topics:

    • Fourier models/time series data (Monday)
    • Priors (Wednesday)
    • Laplace rule of succesion from Bayesian perspective (Wednesday)
    • Relationship between bayesian linear regression and regularization
    • The kernel trick, other topics?

    Reading/notes:

    • Wednesday (11/15) CLASS
    • ER:
      • 7.1 (Priors and posterior) -- Focus on example 7.1.1 and try following/reproducing the calculations with alpha=beta=1 (so the beta distribution, which we haven't discussed, becomes a uniform distribution which you are very familiar with).
      • 10.3.3 (Bayesian linear regression) -- Optional
    • ISLP: Section 6.2, subsection titled "Bayesian Interpretation of Ridge Regression and the Lasso"

    Additional resources

    Assignments due:

    Review and FINAL EXAM

    Review:

    • [TBD]