Math Ladder · College

Probability

Probability with calculus: conditioning, discrete and continuous distributions, joint distributions, covariance, the limit theorems and Markov chains.

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40 topics in 7 units. Builds on Calculus 2: Integrals and Series and Probability and Statistics and Calculus 3: Multivariable Calculus. Not sure this is the right course? Find your rung.

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Unit 1 Probability Spaces

3 topics

Build probability from three rules, and count carefully when outcomes are equally likely.

  1. The Axioms of Probability
  2. Inclusion–Exclusion for Events
  3. Products of Chances: The Birthday Problem

Unit 2 Conditioning

4 topics

Tell joint from conditional probability, split problems into cases, update with Bayes' theorem, and test independence.

  1. Conditional or Joint?
  2. The Law of Total Probability
  3. Bayes' Theorem with Several Causes
  4. Independence of Several Events

Unit 3 Discrete Random Variables

9 topics

Work with pmfs and CDFs, expectations of functions, variance rules and indicators, and the classic discrete distributions.

  1. PMFs and CDFs
  2. Expected Value of a Function
  3. Variance: The Shortcut and the Rules
  4. Linearity of Expectation and Indicators
  5. Sampling without Replacement: The Hypergeometric Distribution
  6. Waiting for the rth Success
  7. The Poisson Distribution
  8. The Poisson Approximation to the Binomial
  9. Choosing a Distribution

Unit 4 Continuous Random Variables

6 topics

Find probabilities, CDFs, means and variances from densities, meet the exponential distribution and the Poisson process, and transform variables.

  1. Probability Density Functions
  2. CDFs, Medians and Percentiles
  3. Mean and Variance of a Continuous Variable
  4. The Exponential Distribution
  5. The Poisson Process
  6. Densities of Functions of a Variable

Unit 5 Joint Distributions

10 topics

Handle two random variables at once: joint tables and densities, conditional distributions, covariance, sums and conditional expectation.

  1. Joint and Marginal Distributions
  2. Conditional Distributions and Independent Variables
  3. Uniform Points and Geometric Probability
  4. Joint Densities
  5. Marginal and Conditional Densities
  6. Covariance and Correlation
  7. Variance of a Sum
  8. Sums of Independent Variables
  9. Conditional Expectation
  10. Maximum and Minimum of Independent Variables

Unit 6 Inequalities and Limit Theorems

5 topics

Bound probabilities with Markov and Chebyshev, then see why averages settle down and sums look normal.

  1. Markov's and Chebyshev's Inequalities
  2. The Law of Large Numbers
  3. Moment Generating Functions
  4. The Central Limit Theorem for Sums
  5. The Normal Approximation to the Binomial

Unit 7 Markov Chains

3 topics

Follow a process that moves between states at random: n-step probabilities, absorption and the long run.

  1. Markov Chains and Transition Matrices
  2. Absorption and First-Step Analysis
  3. Stationary Distributions
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