Topics for Test 1

1Basics of Probability

  • Joint distributions of discrete RVs; Marginals, Conditionals
  • Sum and Product Rules
  • Bayes Theorem
  • Independence

2Discrete Random Variables

  • Expected Value & Variance
  • Bernoulli & Binomial RVs
  • Poisson RV
  • Categorical and Multinomial RVs

3Probability Densities

  • PDFs and CDFs
  • Expectations and Variances
  • Uniform RVs
  • Exponential RVs
  • Gaussian RVs
  • Central Limit Theorem

4Partial Derivatives

  • First order partial derivatives
  • Tangent surfaces
  • Higher-order partial derivatives

5Chain Rule

  • One independent variable
  • Two independent variables
  • Tree diagram of a chain rule

6Gradient & Directional Derivatives

  • Gradient, level hypersurfaces & tangent hyperplanes
  • Directional derivatives
  • Geometric properties of the gradient vector
  • Gradient descent

7Differentials and Jacobians

  • Linear approximations via differentials
  • Jacobians
  • Compositions of transformations and of Jacobians

8Multivariate Optimization

  • Critical and singular points
  • Classifying critical points; second derivatives test
  • The method of Lagrange multipliers
Practice Test 1