Probability
Distributions, expectation, conditional expectation and the limit theorems.
5 lessons open · 26 planned
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Monty Hall and conditional probability
When you learn that something happened, how should a probability change?
PMF, PDF and CDF
A random quantity can be described by a list of probabilities, by a density curve, or by a running total. How do the three fit together?
Expectation and variance
Put each value of a random variable on a number line and give it its probability as a weight. Where does the line balance, and how do you measure how spread out the weight is?
The exponential distribution and memorylessness
You have already waited 10 minutes for a bus. Is the chance that it comes in the next 5 minutes higher, the same, or lower than the chance that it would have come in the first 5 minutes?
From binomial to Poisson
Many small loans that each might default, thousands of characters on a page that each might be a typo, many requests to a server that each might fail: when there are many independent chances and each one is rare, how is the number of events distributed?
Planned
In learning order. These are not pages yet.
- Counting outcomes: permutations, combinations, and the binomial coefficientIntro
- Joint distributions: same marginals, different dependenceAdvanced undergrad
- Quantiles and empirical distributions: inverting a CDF with jumpsIntro
- How a density changes when you transform a random variableIntro
- Covariance and correlation: what ρ = 0.5 looks likeIntro
- The normal distribution and its tail areasIntro
- Moment generating functions: every moment in one functionIntro
- Markov and Chebyshev inequalitiesIntro
- Jensen's inequality: averaging before or after a convex functionIntro
- Survival functions and hazard ratesAdvanced undergrad
- Convolution: the distribution of a sum of independent random variablesIntro
- The lognormal distribution: why the mean sits above the medianAdvanced undergrad
- Characteristic functions: the transform that always existsGraduate
- The law of large numbers: why averages settle downIntro
- Conditional expectation as the best guess given what you knowAdvanced undergrad
- Distributions built from normals: chi-square, t, and FAdvanced undergrad
- Chernoff bounds and large deviations: how fast rare events become rarerGraduate
- The multivariate normal and its contour ellipsesAdvanced undergrad
- The central limit theorem: why sums become normalIntro
- Modes of convergence: almost surely, in probability, in L², in distributionAdvanced undergrad
- Probability spaces and σ-algebras: modeling what you knowAdvanced undergrad
- Change of measure: reweighting outcomes with a likelihood ratioGraduate
- Heavy tails: when the variance doesn't existAdvanced undergrad
- Extreme value theory: the distribution of the maximumGraduate
- Copulas: separating the marginals from the dependenceGraduate
- Uniform integrability: when convergence doesn't carry the expectation alongGraduate
How it connects
Builds on Calculus Real analysis Linear algebra Complex analysis Proof, logic and sets connection
Leads to Stochastic processes Market risk Statistical inference Computational statistics Equities and portfolios Derivatives Regression Arbitrage and risk-neutral pricing Numerical methods Credit risk Linear algebra Differential equations Optimization Stochastic calculus and SDEs Time series Bayesian statistics Corporate finance Bonds and interest rates PDE methods Monte Carlo and numerical pricing
Subjects marked “connection” are linked by a related idea, not by a prerequisite.