Lecture Notes: Probability and Random Processes at KTH for ...

1 Lecture Notes: Probability and Random Processes at KTHforsf2940 Probability TheoryEdition: 2017 Timo KoskiDepartment of MathematicsKTH Royal Institute of TechnologyStockholm, Sweden2 ContentsForeword91 Probability Spaces and Random Introduction .. Terminology and Notations in Elementary Set Theory .. Algebras of Sets .. Probability Space .. Probability Measures .. Continuity from below and Continuity from above .. Why Do We Need Sigma-Fields? .. Negligible Events andP-Almost Sure Properties .. Random Variables and Distribution Functions .. Randomness? .. Random Variables and Sigma Fields Generated by Random Variables.. Distribution Functions .. Independence of Random Variables and Sigma Fields, s .. The Borel-Cantelli Lemmas .. Expected Value of a Random Variable .. A First Definition and Some Developments .. The General Definition.

2 The Law of the Unconscious Statistician .. Three Inequalities for Expectations .. Limits and Integrals .. Appendix: lim supxnand lim infxn.. Sequences of real numbers .. lim supxn.. lim infxn.. Properties, The Limit of a Sequence .. Appendix: lim supAnand lim infAn.. Appendix: Combinatorics of Counting and Statistics of Particlesin Cells .. Exercises .. Easy Drills .. Measures, Algebras and Sigma Fields .. Random Variables and Expectation .. 4434 CONTENTS2 Probability Introduction .. Continuous Distributions .. Univariate Continuous Distributions .. Continuous Bivariate Distributions .. Mean, Variance and Covariance of Linear Combinations of .. Discrete Distributions .. Univariate .. Bivariate Discrete Distributions .. Transformations of Continuous Distributions .. The Probability Density of a Function of Random Variable.

3 Change of Variable in a Joint Probability Density .. Appendix: Decompositions of Probability Measures on the Real Line .. Introduction .. Decompositions of on (R,B) .. Continuous, Discrete and Singular Random Variables .. Exercises .. Distribution Functions .. Univariate Probability Density Functions .. Multivariate s .. Expectations and Variances .. Additional Exercises .. 883 Conditional Probability and Expectation a Sigma Introduction .. Conditional Probability Densities and Conditional Expectations .. Conditioning an Event .. Conditioning a Partition .. Conditioning a Random Variable .. A Case with an Explicit Rule for Conditional Expectation .. Conditioning a -Field .. Properties of Conditional Expectation .. An Application of the Properties of Conditional Expectation a -Field.

4 Estimation Theory .. Tower Property and Estimation Theory .. Jensen s Inequality for Conditional Expectation .. Exercises .. Easy Drills .. Conditional Probability .. Joint Distributions & Conditional Expectations .. Miscellaneous .. Martingales .. 112 CONTENTS54 Characteristic On Transforms of Functions .. Characteristic Functions: Definition and Examples .. Definition and Necessary Properties of Characteristic Functions .. Examples of Characteristic Functions .. Characteristic Functions and Moments of Random Variables .. Characteristic Functions of Sums of Independent Random Variables .. Expansions of Characteristic Functions .. Expansions and Error Bounds .. A Scaled Sum of Standardised Random Variables (Central Limit Theorem) .. An Appendix: A Limit .. A Sequence of Numbers with the Limitex.. Some Auxiliary Inequalities.

5 Applications .. Exercises .. Additional Examples of Characteristic Functions .. Selected Exam Questions from the Past Decades .. Various Applications of the Characteristic Function .. Mellin Transform in Probability .. 1395 Generating Functions in Introduction .. Probability Generating Functions .. Moments and Probability Generating Functions .. Probability Generating Functions for Sums of Independent Random Variables .. Sums of a Random Number of Independent Random Variables .. The Probability of an Even Number of Successes .. Moment Generating Functions .. Definition and First Properties .. is really an Exponential Moment Generating Function, ! .. Exercises .. Probability Generating Functions .. Moment Generating Functions .. Sums of a Random Number of Independent Random Variables .. Various Additional Generating Functions in Probability .

6 The Chernoff Inequality .. 1646 Convergence of Sequences of Random Introduction .. Definitions of Modes of Convergence, Uniqueness of the Limit .. Relations between Convergences .. Some Rules of Computation .. Asymptotic Moments and Propagation of Error .. Convergence by Transforms .. Theorems on Convergence by Characteristic Functions .. Convergence and Generating Functions .. Central Limit Theorem .. Almost Sure Convergence .. Definition .. Almost Sure Convergence Implies Convergence in Probability .. A Summary of the General Implications between Convergence Concepts and One SpecialImplication .. The Strong Law of Large Numbers .. Exercises .. Convergence in Distribution .. Central Limit Theorem .. Convergence in Probability .. Proof of Theorem .. Almost Sure Convergence, The Interrelationship Between Almost Sure Convergence andMean Square Convergence, Criteria for Almost Sure Convergence.

7 1907 Convergence in Mean Square and a Hilbert Convergence in Mean Square; Basic Points of View .. Definition .. The Hilbert SpaceL2( ,F,P) .. Cauchy-Schwartz and Triangle Inequalities .. Properties of Mean Square Convergence .. Applications .. Mean Ergodic Theorem .. Mean Square Convergence of Sums .. Mean Square Convergence of Normal Random Variables .. Subspaces, Orthogonality and Projections inL2( ,F,P) .. Exercises .. Mean Square Convergence .. Optimal Estimation as Projection on Closed Linear Subspaces inL2( ,F,P) .. 2018 Gaussian Multivariate Gaussian Distribution .. Why Gaussianity ? .. Notation for Vectors, Mean Vector, Covariance Matrix & Characteristic Functions .. Multivariate Normal/Gaussian Distribution .. Partitioned Covariance Matrices .. Appendix: Symmetric Matrices & Orthogonal Diagonalization & Gaussian Vectors.

8 Appendix: Proof of ( ) .. Exercises .. Bivariate Gaussian Variables .. Covariance Matrices & The Four Product Rule .. Bussgang s Theorem & Price s Theorem .. 223 CONTENTS79 Stochastic Processes : Weakly Stationary and Stochastic Processes .. Definition and Terminology, Consistency Theorem .. Mean Function, Autocorrelation Function .. Mean Square Calculus: The Mean Square Integral .. Definition and Existence of the Mean Square Integral .. Weakly Stationary Processes .. Bochner s Theorem, Spectral Density and Examples of Autocorrelation Functions forWeakly Stationary Processes .. Mean Square Continuity of Weakly Stationary Processes .. Gaussian Processes .. Definition and Existence .. Weakly Stationary Gaussian Processes .. Distribution of Mean Square Integrals of Gaussian Processes .. The Gauss-Markov Processes and Separable Autocorrelation Functions.

9 The Markov Property Defined and Characterized .. The Chapman-Kolmogorov (or Smoluchowski) Equation for Transition Densities .. Gauss-Markov Processes and Separable Autocorrelation Functions .. What Can NotBe Computed by the Methods Introduced Above ? .. Exercises .. Autocovariances and Autocorrelations .. Examples of Stochastic Processes .. Autocorrelation Functions .. Weakly Stationary Processes .. Gaussian Stationary Processes .. Mean Square Integrals of Processes .. Mean Square Continuity .. Memoryless Nonlinear Transformations of Gaussian Processes .. Separable Autocovariances .. 26410 The Wiener Introduction .. Background: The Brownian Movement, A. Einstein .. Diffusion, Theory of Speculation & the Wiener Process .. The Wiener Process: Definition and First Properties .. A Construction of the Wiener Process .. The Sample Paths of the Wiener Process.

10 The Sample Paths of the Wiener Process are Almost Surely Continuous .. The Sample Paths of the Wiener Process are Almost Surely Nowhere Differentiable;Quadratic Variation of the Sample Paths .. The Wiener Process is a Markov Process .. The Wiener Integral .. Definition .. Properties .. The Wiener Integral is a Scrambled Wiener Process .. White Noise .. Martingales and the Wiener Process .. Exercises .. Random Walks .. Wiener Process .. The Wiener Integral .. A Next Step: Stochastic Calculus .. 29811 The Langevin Equations and the Ornstein-Uhlenbeck On Langevin Equations .. The Ornstein-Uhlenbeck Process .. Mean-Squared Displacement: The Langevin Theory .. The Langevin Equations for Thermal or Nyquist- Johnson Noise.. Exercises .. 30912 The Poisson Introduction .. Definition and First Properties .. The Counter Process and the Poisson Process.