An Introduction to Probabilistic ModelingIntroduction to the basic concepts of probability theory: independence, expectation, convergence in law and almost-sure convergence. Short expositions of more advanced topics such as Markov Chains, Stochastic Processes, Bayesian Decision Theory and Information Theory. |
Contents
Conditional Probability and Independence | 12 |
Concrete Probability Spaces | 26 |
CHAPTER 2 | 41 |
Variance and Chebyshevs Inequality | 56 |
GaltonWatsons | 64 |
CHAPTER 3 | 77 |
Probability Densities | 85 |
Expectation of Functionals of Random Vectors | 96 |
CHAPTER 4 | 128 |
Poisson Processes | 140 |
Gaussian Stochastic Processes | 146 |
Tests | 148 |
CHAPTER 5 | 163 |
The ChiSquare Test | 180 |
Additional Exercises | 193 |
Solutions to Additional Exercises | 199 |
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Common terms and phrases
A₁ admits almost-sure convergence axioms B₁ central limit theorem Chapter characteristic function Chebyshev's inequality coin compute convergence in law counting process covariance matrix cumulative distribution function defined definition denoted density fx discrete random variable equality EXAMPLE Exercise Figure finite formula Gaussian random variable Gaussian vector independent random variables integral k₁ large numbers law of large Lebesgue linearity m₁ mathematical N₁ N₂ nonnegative notation obtained P-as P(X₁ P(Y₁ P₁ parameter Poisson process prefix code probabilistic model probability density probability space Probability Theory PROOF OF EQ quadratic mean random element random vector real random variables result S₁ sample sequence of random Show stochastic process strong law subsets Suppose t₁ taking its values U₁ variance o² X₁ X₂ Y₁ Y₂ Z₁ σ²