S29 Random Processes (Parts 1 - 7)
with Dr. Harry L. Van Trees
48 videotape set
English
$8,745
individual videotape $400 ea.

This series is a follow-on to S28 Probability and a prerequisite to study in such areas as advanced communications and queuing theory. Included are discussions of three completely characterized processes – the Poisson, Markov, and Gaussian processes.

The complete course includes: 48 black & white videotapes; a complete set of seven study guides in five volumes with chalkboard photographs, reading assignments, problem sets, solutions, and quizzes; and one textbook –

"Probability and Random Processes". One set study guides is recommended for each participant, and may be purchased for $115.00 each, plus shipping and handling. One Textbook is also recommended for each participant and may be purchased for $52.95 each, plus shipping and handling.

Part I – Introduction to Random Processes

Part II – Linear Systems

Part III – Second Moment Theory

Part IV – Poisson Processes

Part V – Markov Processes

Part VI – Gaussian Processes

Part VII – Measurement of Process Characteristics

Please see individual videotape descriptions below for detailed information.

S29A  Random Processes
Part I: Introduction to Random Processes
with Dr. Harry L. Van Trees
7 videotapes
English
$1,295

Tape One – Introduction to Random Processes

· Introduces, through the use of examples, the analytical approaches by which random process theory
may be applied to a variety of physical problems.

Tape Two – Random Processes: Basic Concepts and Definitions

· Extends the usual sample space definition of random variables to the case of random processes, and|
thereby introduces the notion of a complete characterization.

Tape Three – Fixed-Form Random Processes

· Discusses the simple case of fixed-form processes, wherein a small number of random variables
completely characterize the process.

Tape Four – Binary Transmission Wave

· Treats the binary transmission wave and introduces stationarity concepts.

Tape Five – Random Telegraph Wave

· Considers the random telegraph wave and uses that example to introduce the minimum mean-square error (MMSE) prediction problem.

Tape Six -

Tape Seven -

S29B Part II Random Processes:
Linear Systems
with Dr. Harry L. Van Trees

7 videotapes
$1,095.00

Tape One -

Tape Two -

Discusses the characterization of a linear time-invariant (LTI) system in the time domain by its

impulse response.

Tape Three – Measurement of Impulse Response

· Demonstration of the relative invariance of linear time-invariant system responses to the detailed shape

of a pulse input of short duration but fixed area.

Tape Four – Convolution Integral

· Provides a further investigation of the classification of systems according to their input-output properties.

Tape Five – System Classification

· Provides a further investigation of the classification of systems according to their input-output properties.

Tape Six – Complex Exponential Inputs: Frequency Domain Analysis

· Begins the frequency domain analysis of LTI systems by considering the response of such systems to

complex exponential excitations.

Tape Seven – Periodic Inputs and Fourier Series

· Continues the frequency domain analysis of LTI systems by developing the Fourier series representation for

periodic signals and considering the response of such systems to periodic inputs.

Tape Eight – Fourier Series Demonstration

· Approximation representation of a square wave by a sum of sine and co-sine waves –

even and odd functions, even and odd harmonics, Gibb’s phenomenon, mean square error property.

Tape Nine – Fourier Transforms

· Makes the transition from the Fourier series to the Fourier Transform representation for aperiodic signals,

and derives the convolution-multiplication theorem that relates the time and frequency domain

representations of LTI system input-output pairs.

Tape Ten – System Functions

· Describes some techniques for measuring system functions (impulse response and frequency response), and

also considers the analysis of linear systems characterized by differential equations, or the cascade of several

linear systems.

Tape Eleven – Fourier Transform Properties

· Treats the mathematical properties of the Fourier transform.

Tape Twelve – Sampling Theorem

· Discusses the presentation of a bandlimited waveform by its time samples.

S29C Random Processes Part III:
Second Moment Theory
with Dr. Harry L. Van Trees

8 videotapes
English
$1,475

Tape One – Linear Systems with Random Process Inputs

· Begins the study of linear filtering of random processes by deriving the mean function and correlation

function of the output of a linear system driven by noise.

Tape Two – Time Averages

· Builds upon the results of the previous lecture to investigate the relationship between statistical

(ensemble) averages and empirical (time) averages of random processes.

Tape Three – Frequency Domain Analysis of Stationary Random Processes

· Develops the basic frequency domain analysis of wide-sense stationary random processes.

Tape Four – White Noise

· Devoted to the definition and use of white-noise processes in linear-system calculations.

Tape Five – Two Applications of White Noise

· Discusses two applications of white noise: synthesis of a random process with a desired spectrum,

and measurement of linear-system impulse response.

Tape Six – Matched Filters

· Derives the matched-filter as the optimum linear processor for detection of a known signal in additive

white noise.

Tape Seven – Optimum Fixed-Form Linear Filters

· Continues the discussion of linear signal processing by considering optimum fixed-form filters for estimating

a random signal embedded in additive noise.

Tape Eight – Optimum Linear Filters

· Concludes the discussion of linear signal processing by deriving the optimum unrealizable filter for estimating

a random signal embedded in additive noise.

S29D Random Processes Part IV:
Poisson Processes
with Dr. Harry L. Van Trees

5 videotapes
English
$970

Tape One – Introduction to Poisson Processes

· Introduces the Poisson counting process through its independent increments property and mentions its
potential application areas.

Tape Two – Poisson Counting Processes

· Reviews the definition of the Poisson counting process and calculates the counting probabilities from its incremental statistics.

Tape Three – Arrival Times

· Derives the arrival-time statistics for the Poisson counting process.

Tape Four – Filtered Poisson Process

· Builds upon previous results to study the statistics of a linearly-filtered Poisson impulse train – i.e., a shot- noise process.

Tape Five – Limiting Behavior of Filtered Poisson Processes

· Continues the development of shot-noise statistics by demonstrating the approach of high-density shot-noise to a Gaussian distribution.

S29E Random Processes Part V:
Markov Processes
with Dr. Harry L. Van Trees

6 videotapes
English
$1,095 

Tape One – Introduction to Markov Processes

· Begins the study of discrete-state continuous-time Markov processes by use of examples, and introduces the state transition diagram.

Tape Two – Markov Process Equations

· Considers basic analysis techniques for finding the transient and equilibrium statistics of a discrete-state continuous-time Markov process.

Tape Three – Finite-State Processes

· Discusses explicit solution techniques for the transient and equilibrium behavior of finite-state Markov processes.

Tape Four – Pure Birth Process

· Obtains the state-occupation probabilities for a pure-birth process.

Tape Five – Linear Birth Process

· Continues the discussion of the previous lecture for the case of linear birth processes.

Tape Six – Equilibrium Distributions: Infinite-State Processes

· Investigates the existence of and solution for the equilibrium distribution in an infinite-state Markov process.

S29F Random Processes Part VI:
Gaussian Processes
with Dr. Harry L. Van Trees

7 videotapes
English
$1,295

Tape One – Introduction to Gaussian Random Processes

· Introduces the Gaussian process through examples that make use of the central limit theorem.

Tape Two – Gaussian Random Vectors

· Develops the properties of Gaussian random vectors including joint probability density, and characteristic functions.

Tape Three – Gaussian Random Processes

· Defines the Gaussian random process in terms of Gaussian random vectors and uses the definition to show that the mean and covariance function completely characterize the Gaussian process.

Tape Four – Gaussian Processes and Linear Systems

· Combines the results of the previous lecture and those of second moment theory to completely characterize the output of a linear system driven by the Gaussian noise.

Tape Five – Gaussian Processes and Nonlinear Systems

· Uses the moment factoring property of Gaussian random variables to investigate the response of nonlinear systems to Gaussian noise.

Tape Six – Linear Optimality and General Optimality

· Uses the results of previous lectures in conjunction with the optimum linear filtering results of second moment theory to show that linear optimality is identical to global optimality for Gaussian processes.

Tape Seven – Summary of Gaussian Processes

· Concludes the study of Gaussian processes by summarizing the key properties derived in the previous lectures.

S29G Random Processes Part VII:
Measurement of Process Characteristics
with Dr. Harry L. Van Trees

3 videotapes
English
$580

Tape One – Measurement of Random Process Characteristics

· Introduces the goals of and problems encountered in measuring the statistics of a random process from sample-function observations.

Tape Two – Measurement of Mean-Square Value of a Random Process

· Evaluates the bias and variance of the time-average estimate of the mean-square value of a Gaussian random process.

Tape Three – Measurement of Power Density Spectra

· Provides an introduction to spectral density estimation and develops an appreciation for the tradeoff between resolution and accuracy.

S28 Probability (Parts 1 - 4)
with Dr. Harry L. Van Trees
49 videotape set
English
$8,950
individual videotape $400 ea.

This series is a post-calculus approach to this important mathematical discipline. Even a cursory survey of engineering, for example, reveals the widespread applicability of probability theory. Such diverse fields as systems analysis, decision theory, statistics, automatic control, modern management, and cybernetics all rely on a probabilistic approach. S29 Random Processes is a follow-on to Probability.

The complete course includes: 49 black & white videotapes; a complete set of four study guides with summaries of lecture concepts, reading assignments, problems, quizzes, and problem solutions; one textbook – "Probability and Random Processes"; one set of Lecture Notes which includes chalkboard photographs from all the videotapes; and one Mathematics Pretest to determine proficiency in calculus concepts and techniques used in Probability. One set of study guides is recommended for each participant, and may be purchased for $35.00 each, plus shipping and handling. One Textbook is also recommended for each participant and may be purchased for $52.95 each, plus shipping and handling. One set of Lecture Notes is recommended for each participant and may be purchased for $5.00 each, plus shipping and handling.

Part I – Elementary Probability Theory

Part II – Random Variables

Part III – Statistical Averages

Part IV – Limit Theorems and Statistics

Please see individual videotape descriptions below for detailed information.

S28A Probability Part I: Elementary Probability Theory
with Dr. Harry L. Van Trees
13 videotapes
English
$2,495

Tape One – Introduction to Probability

· This lecture presents a few examples of cases where probability theory is applied.

Tape Two – Formulation of Mathematical Models (1)

· Formulation of mathematical models in probability theory – deals with the definition of a probabilistic experiment and with the definition of an event.

Tape Three – Formulation of Mathematical Models (2)

· Continues discussion of formulation of mathematical models – events are defined as a collection of sample points.

Tape Four – Elementary Set Theory

· Introduces basic ideas of elementary set theory. Defines and illustrates graphically the ideas of equality, inclusion, union, intersection, complementarily, difference, null-sets, disjoint sets, and partitioning.

Tape Five – Theorem Proving

· Introductory lecture on theorem proving – Proof of an "IF and only IF" type of theorem – Proof by contradiction.

Tape Six – Probabilistic Models

· Covers the five basic axioms of probability theory and illustrates those axioms through a number of examples.

Tape Seven – Proof by Induction

· Continuation of theorem proving – Illustrates in detail proof by induction and presents an example involving the two basic steps of typical proof by induction.

Tape Eight – Joint Probability

· Idea of joint probability is introduced through the use of two examples – Shows that joint probabilities must obey the axioms of probability theory.

Tape Nine – Conditional Probability (1)

· Illustrates how probabilities change when events are conditioned by other events.

Tape Ten – Conditional Probability (2)

· Continuation of the previous lecture – The lecture goes through a detailed example on reliability to illustrate the ideas of conditional probability – Bayes rule is derived.

Tape Eleven – Conditional Probability: A Digital Communications Example

· Construction of sample space from conditional probability assumptions or measurements – Application of probabilistic ideas to the design of the system.

Tape Twelve – Statistical Independence

· The fundamental concept of statistical independence is defined and its meaning is illustrated through a number of examples. The utility of this concept in probabilistic analysis is discussed briefly.

Tape Thirteen – Product Spaces and Statistically Independent Experiments

· Extends the concept of statistical independence – Extends the concept of statistical independence – Construction of product spaces for statistically independent experimental outcomes – Successive coin tosses.

S28B Probability Part II: Random Variables
with Dr. Harry L. Van Trees
16 videotapes
English $2,990

 Tape One – Random Variables (1)

· Introduction to random variables – Probability distributions and probability distribution functions are defined – Properties of probability distribution functions.

Tape Two – Random Variables (2)

· Continuous random variables – Probability density functions and their properties – Example of a uniform random variable.

Tape Three – Canonical Random Variables

· Describes the standard variables often used in practice – Exponential random variable is introduced and illustrated.

Tape Four – Mixed Random Variables

· Random variables are classified as continuous, discrete or mixed – The definition of an impulse is provided – Detailed example illustrates mixed random variables.

Tape Five – Conditioning

· The conditioning of random variables is defined. It is shown that conditional random variables have all the properties of ordinary random variables.

Tape Six – Multiple Random Variables: Discrete

· Definition of joint probability distribution functions – Definition of marginal probability distribution functions    
– Discussion of the properties of both with a detailed example presented.

Tape Seven – Continuous Random Variables

· Joint probability distribution functions for continuous random variables – Joint probability density functions, marginal distribution functions, and marginal density functions for continuous random variables.

Tape Eight – Impulsive Densities

· Densities containing impulses are discussed. Integration and differentiation for these densities are illustrated.

Tape Nine – Statistically-Independent Random Variables

· Statistical independence is defined in terms of probability distribution functions. The concept is illustrated with the derivation of a marginal density function. The idea of utility is introduced.

Tape Ten – Conditioning by Sets

· The conditioning of probability distributions and probability densities on sets is illustrated – Conditional distribution functions are defined.

Tape Eleven – Point Conditioning

· Conditioning is extended to point-conditioning. This leads to the definition of conditional distribution functions of random variables.

Tape Twelve – A Digital Communication Application

· A detailed example of a communication system with noise added is presented. Model of a communication system and the idea of minimum error decision – Computation of error probabilities.

Tape Thirteen – Functions of a Random Variable

· Computation of probability distribution for functions of a single random variable – Standard procedure for this computation with illustrations.

Tape Fourteen – Functions of Vector Random Variables (1)

· Computation of probability distributions for functions of vectors of random variables – Standard procedure – Special case of statistically independent random variables.

Tape Fifteen – Reliability Applications

· Introduction to the computation of reliability – Standard configurations of networks – Components in series and in parallel.

Tape Sixteen – Functions of Vector Random Variables (2)

· More complicated derivations of probability distributions for functions of vectors of random variables with illustration through example.

S28C Probability Part III: Statistical Averages
with Dr. Harry L. Van Trees
11 videotapes
English
$2,095

Tape One – Statistical Averages: Expectation of a Random Variable

· The fundamental concept of expectation is introduced – Computation of expected values for continuous and discrete random variables.

Tape Two – Expectations of Functions of a Random Variable

· The concept of expectation is extended to functions of random variables – Examples are presented.

Tape Three – Moments of a Random Variable

· Moments and central moments are defined – Variance and standard deviation – Properties of variance and examples.

Tape Four – The Chebyshev Inequality

· The Chebyshev inequality is derived and explained. The lecture also includes a discussion of how good a Chebyshev inequality is as a bound on probabilities

Tape Five – Estimation of Random Variables

· Deals with the choice of estimators for random variables – The mean-square error is discussed as a criterion for estimation with two examples.

Tape Six – Conditional Expectation

· The idea of conditional expectation is introduced – Conditional expectation on a joint Gaussian probability density plus the importance of the concept in estimation.

Tape Seven – Minimum Mean-Square Error Estimation

· Reviews the idea of the minimum mean-square error estimation with an example involving the joint Gaussian probability density.

Tape Eight – Joint Moments: Correlation

· Joint moments of random variables are defined. Correlation covariance and the correlation coefficients – Basic properties of joint moments plus predictive values.

Tape Nine – Linear Estimation

· Linear estimation is introduced. Fundamental expressions are derived – Minimum mean-square estimators are discussed in this light.

Tape Ten – Characteristic Functions

· Characteristic functions are defined and the Four fundamental properties of characteristic functions are derived.

Tape Eleven – Joint Characteristic Functions

· The concept of a characteristic function is extended to vectors of random variables – Properties of joint characteristic functions are derived and discussed.

S28D Probability Part IV: Limit Theorems and Statistics
with Dr. Harry L. Van Trees

9 videotapes
English
$1,695

Tape One – Sample Means and the Weak Law of Large Numbers

· The sample mean is presented as an estimator of expectation – The weak law of large numbers –
Convergence in the mean-square sense – Discussion of different types of convergence.

Tape Two – Relative Frequency

· Relative frequency is defined – Shows that the relative frequency of an event converges to the probability of that event – An example is presented.

Tape Three – The Gaussian Approximation

· Introduces the idea of using the Gaussian approximation for large samples. An example using binomial
distribution is discussed in detail with comparison between exact estimates and the Gaussian approximation.

Tape Four – Central Limit Theorem

· The derivation of the central limit theorem is outlined. Its implications are discussed in detail –Illustration approximations through the central limit theorem.

Tape Five – Introduction to Statistical Inference

· Estimation of a probability density, estimation of moments, hypothesis testing, testing with unspecified
alternatives – The meaning of statistics.

Tape Six – Estimation of the Moments of a Random Variable

· Estimation of moments – Unbiased estimators – Consistent estimators – Normalized variance and Illustration through the use of two statistical problems.

Tape Seven – Estimation of the Parameter of a Probability Density

· Procedure and issues in the estimation of parameters of probability densities – Likelihood functions – Maximum likelihood estimators and Biases.

Tape Eight – Performance Bounds: The Cramer-Rao Inequality

· The Cramer-Rao inequality as a lower bound on the variance of unbiased estimators – Efficient estimators and comments on the use of efficient estimators.  

Tape Nine – Estimation of the Probability Density of a Random Variable

· Detailed example on the estimation of a probability density function. Pitfalls and procedures for the estimation of probability densities and the use of variance analysis.