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Advanced Probability

The deepest layer of the probability track: how a single uniform generates every distribution, why the normal is inevitable, the moment-generating function as a distribution's fingerprint, multivariate probability and covariance, the inequalities that bound the unknown, the limit theorems at the heart of statistics, and Markov chains. Intuition first, with every derivation intact.

Rapid Recall

The universality of the uniform says any distribution is a warped uniform via its inverse CDF, which is why one RNG simulates anything. The normal's bell shape is forced by symmetry and decay, with its \(\sqrt{2\pi}\) constant pinned by the Gaussian integral. The MGF \(E(e^{tX})\) stores every moment and turns independent sums into products. Joint, marginal, and conditional distributions are one landscape seen three ways; covariance measures linear co-movement and correlation standardizes it. The inequalities (Cauchy-Schwarz, Jensen, Markov, Chebyshev) bound probabilities under partial knowledge. The law of large numbers gives the destination and the central limit theorem gives the shape of the approach. Markov chains hop between states where only the present matters, and their stationary distribution is a left eigenvector.

What this section covers

The arc of the section

flowchart LR
  Uniform["Universality<br/>of the uniform"] --> Normal["The Normal"]
  Normal --> MGF["MGFs"]
  MGF --> CLT["Central Limit Theorem"]
  Joint["Joint & conditional"] --> Cov["Covariance & correlation"]
  Cov --> CondE["Conditional expectation"]
  Ineq["Inequalities"] --> LLN["Law of Large Numbers"]
  LLN --> CLT
  CondE --> Markov["Markov chains"]