🎲Stat 110: Probability

Sample spaces, every way there is to count something, and the two moves — inclusion-exclusion and story proofs — that tame hard counting problems.

1.1 Sample Spaces and the Naive Definition of Probability · 1.1 The Multiplication Rule for Counting · 1.2 Permutations: Counting Orderings · 1.2 Combinations and the Binomial Coefficient · +9 more

What happens once you learn something is true — the sample space shrinks, Bayes' rule flips the conditioning, and independence is stronger than 'unrelated'.

2.1 Defining Conditional Probability · 2.2 Bayes' Rule · 2.2 Bayes' Rule as a Tree of Natural Frequencies · 2.3 The Law of Total Probability · +8 more

Turning outcomes into numbers, and the discrete distribution family — Bernoulli through Poisson — that a huge share of real-world randomness reduces to.

3.1 What a Random Variable Actually Is · 3.2 PMFs and CDFs · 3.2 The Discrete Uniform Distribution · 3.3 The Bernoulli Distribution · +7 more

The single number that summarizes a random variable's center of mass — built to stay linear no matter how tangled the dependence underneath it gets.

4.1 Defining Expected Value · 4.2 Linearity of Expectation · 4.2 The Coupon Collector Problem · 4.2 The St. Petersburg Paradox · +4 more

Probability spread over a continuum instead of piled on points — PDFs, the Uniform's universality, the Normal curve, and the Exponential's total lack of memory.

5.1 PDFs: Probability Density Instead of Probability Mass · 5.1 The CDF-PDF Relationship · 5.2 The Uniform Distribution · 5.2 The Universality of the Uniform · +5 more

A single transform that packages every moment of a distribution at once, and turns adding independent random variables into multiplying functions.

6.1 Defining the Moment Generating Function · 6.1 The MGF Uniquely Determines a Distribution · 6.2 Using the MGF to Find Moments · 6.2 MGFs of Sums of Independent Random Variables · +2 more

More than one random variable at once — joint, marginal, and conditional distributions, 2D LOTUS, covariance, and the Multinomial and Multivariate Normal.

7.1 Joint, Marginal, and Conditional Distributions · 7.1 Independence of Random Variables · 7.2 The Chicken-Egg Problem: Poisson Thinning · 7.3 2D LOTUS · +6 more

Turning one random variable into another, sliding one density across another to add them, and the Beta-Gamma pair behind the Poisson process.

8.1 Transformations of a Random Variable · 8.1 The Log-Normal Distribution · 8.2 The Probabilistic Method · 8.2 Convolutions: Adding Two Random Variables · +5 more

Sorting a sample and asking what its extremes look like, then the deepest tool in the course — conditioning a random variable on another random variable.

9.1 Order Statistics · 9.1 The Distribution of the Minimum · 9.2 Conditional Expectation Given a Random Variable · 9.2 Taking Out What's Known · +5 more

Bounds that hold for any distribution at all, and the two theorems that make averages and sums predictable no matter the underlying distribution.

10.1 Markov's Inequality · 10.2 Chebyshev's Inequality · 10.3 The Cauchy-Schwarz and Jensen Inequalities · 10.4 The Law of Large Numbers · +4 more

A process that forgets everything but where it is now — transition matrices, which states keep getting revisited, and the long-run distribution it settles into.

11.1 Defining a Markov Chain · 11.2 Transition Matrices and n-Step Transitions · 11.3 Classifying States: Recurrent, Transient, and Periodic · 11.3 Irreducibility · +5 more

Turning the last unit's convergence guarantee around: build a chain whose stationary distribution IS the hard distribution you actually wanted to sample from.

12.1 Why Sample From a Hard Distribution With a Markov Chain · 12.2 The Metropolis-Hastings Algorithm · 12.3 Sampling a Discrete Distribution With MCMC · 12.4 Where MCMC Shows Up in Practice