Probability Theory | Index

Probability theory is the calculus-based study of chance: random variables, their distributions and what happens to averages over many trials.

The course covers the axioms of probability, discrete and continuous distributions, moment generating functions, joint distributions, covariance and the limit theorems.

It’s for college students in a calculus-based probability course, including math, statistics, engineering and actuarial science majors.

What to know first: probability and random variables in Statistics, plus Calculus 2. Double integrals from Calculus 3, Unit 4 are used in Unit 4.

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: Probability Foundations

  1. Axioms of Probability (coming soon) — the three axioms of probability and the rules that follow from them
  2. Probability with Combinations (coming soon) — find probabilities of card hands, lotteries and committees by counting
  3. Law of Total Probability (coming soon) — find a probability by splitting the sample space into cases

Also see: probability rules, conditional probability and Bayes’ theorem in Statistics, Unit 4, and permutations and combinations (Precalculus).

Unit 2: Discrete Distributions

  1. Poisson Distribution (coming soon) — find probabilities for counts of events in a fixed time or space
  2. Hypergeometric Distribution (coming soon) — find probabilities when sampling without replacement
  3. Negative Binomial Distribution (coming soon) — find the probability that the rth success happens on a given trial

Also see: discrete random variables, expected value and the binomial and geometric distributions in Statistics, Unit 5.

Unit 3: Continuous Random Variables

  1. Probability Density Functions (coming soon) — find probabilities as areas under a density function
  2. Cumulative Distribution Functions (coming soon) — find and use the CDF of a random variable
  3. Expected Value of a Continuous Random Variable (coming soon) — find the mean and variance with integrals
  4. Uniform Distribution (coming soon) — the continuous uniform distribution and its mean and variance
  5. Exponential Distribution (coming soon) — model waiting times and use the memoryless property
  6. Poisson Process (coming soon) — connect Poisson counts and exponential waiting times
  7. Gamma Distribution (coming soon) — the gamma function and the waiting time for several events
  8. Functions of a Random Variable (coming soon) — find the distribution of Y = g(X) with the CDF method

Also see: density curves and the normal distribution in Statistics, Unit 6, and improper integrals (Calculus 2).

Unit 4: Joint Distributions

  1. Joint Probability Distributions (coming soon) — joint probability tables and joint density functions
  2. Marginal and Conditional Distributions (coming soon) — find one variable’s distribution from a joint distribution
  3. Independent Random Variables (coming soon) — test whether two random variables are independent
  4. Covariance (coming soon) — measure how two random variables vary together, and find their correlation
  5. Conditional Expectation (coming soon) — find E(X | Y) and use the law of total expectation
  6. Convolution of Random Variables (coming soon) — find the distribution of a sum of independent random variables
  7. Bivariate Normal Distribution (coming soon) — the joint normal distribution of two correlated variables
  8. Moment Generating Functions (coming soon) — find means and variances, identify distributions and find the distribution of a sum with the MGF

Also see: combining random variables (Statistics) and double integrals in Calculus 3, Unit 4.

Unit 5: Limit Theorems

  1. Markov and Chebyshev Inequalities (coming soon) — bound a probability for a nonnegative variable using its mean, or for any variable using its mean and variance
  2. Law of Large Numbers (coming soon) — prove that sample means settle near the population mean

Also see: the central limit theorem (Statistics).

For Teachers: Standards Alignment

Lessons follow a typical one-semester calculus-based probability course for math, statistics and engineering majors, and cover many topics on the Society of Actuaries Exam P. There is no AP exam for this course. Introductory probability, the binomial, geometric and normal distributions and the central limit theorem are taught in Statistics.

Previous course: Numerical Methods Index