Канал: Probability and Random Processes

Distribution of Minimum of Random Variables

Distribution of Minimum of Random Variables

Sample mean and Sample Vriance for a Normal i. i. d. sequence

Sample mean and Sample Vriance for a Normal i. i. d. sequence

What is a Mean-Ergodic Stochastic Process

What is a Mean-Ergodic Stochastic Process

Example for a Mean-Ergodic Stochastic Process

Example for a Mean-Ergodic Stochastic Process

What is an Ergodic Stochastic Process

What is an Ergodic Stochastic Process

Kosambi–Karhunen–Loève Expansion

Kosambi–Karhunen–Loève Expansion

MS periodic Random Processes

MS periodic Random Processes

Line spectra - example for AR process

Line spectra - example for AR process

Auto-Regressive (AR) processes

Auto-Regressive (AR) processes

Examples - Power spectral factorization

Examples - Power spectral factorization

Solving Yule-Walker equations for AR(1)

Solving Yule-Walker equations for AR(1)

Power spectral factorization of discrete-time RPs using minimum phase filters

Power spectral factorization of discrete-time RPs using minimum phase filters

Moving Average processes

Moving Average processes

Spectral Representation

Spectral Representation

Definition of an Auto Regressive Moving Average Random processes

Definition of an Auto Regressive Moving Average Random processes

Example for Moving Average processes

Example for Moving Average processes

Auto-Regressive Moving Average (ARMA) processes

Auto-Regressive Moving Average (ARMA) processes

Finite-Order Random Processes

Finite-Order Random Processes

Power spectral factorization of rational spectra

Power spectral factorization of rational spectra

Finite-Order Digital LTI Systems

Finite-Order Digital LTI Systems

Power Spectrum Factorization using Rational Polynomials

Power Spectrum Factorization using Rational Polynomials

Spectral Representation Big Picture

Spectral Representation Big Picture

Discrete-time Random Processes

Discrete-time Random Processes

Sampling of Random Processes

Sampling of Random Processes

Sampling Theorem for WSS Signals

Sampling Theorem for WSS Signals

Power Spectral Density of WSS Discrete time Random Processes

Power Spectral Density of WSS Discrete time Random Processes

Example 2 PSD of WSS Discrete time Signals

Example 2 PSD of WSS Discrete time Signals

Example 3 PSD of WSS Discrete time Signals

Example 3 PSD of WSS Discrete time Signals

Example 1 PSD of WSS Discrete time Signals

Example 1 PSD of WSS Discrete time Signals

Discrete and Continuous time Random Processes

Discrete and Continuous time Random Processes

Two functions of two random variables

Two functions of two random variables

Wiener–Khinchin–Einstein Theorem

Wiener–Khinchin–Einstein Theorem

Analysis of Hilbert transform and analytic signal

Analysis of Hilbert transform and analytic signal

Analysis of a differential equation of first order

Analysis of a differential equation of first order

PSD analysis of a moving average filter

PSD analysis of a moving average filter

PSD analysis of Linear Time - Invariant Systems

PSD analysis of Linear Time - Invariant Systems

Doppler effect on PSD of multicomponent sinusoidal

Doppler effect on PSD of multicomponent sinusoidal

Emission and Absorption PSD of atoms

Emission and Absorption PSD of atoms

Doppler effect on PSD

Doppler effect on PSD

PSD and frequency conversion

PSD and frequency conversion

PSD of White noise

PSD of White noise

Example for Analysis of Signals and LTI Systems: Deterministic versus Stochastic

Example for Analysis of Signals and LTI Systems: Deterministic versus Stochastic

Analysis of two jointly WSS RP's

Analysis of two jointly WSS RP's

Analysis of WSS RPs and LTI Systems

Analysis of WSS RPs and LTI Systems

Analysis of Signals and LTI Systems - Deterministic versus Stochastic

Analysis of Signals and LTI Systems - Deterministic versus Stochastic

Power Spectral Densities

Power Spectral Densities

Vector Processes and MIMO Systems

Vector Processes and MIMO Systems

Finite-order linear differential equation-General moments

Finite-order linear differential equation-General moments

Finite-order linear differential equation

Finite-order linear differential equation

Poisson impulses

Poisson impulses

Response of a Differentiator

Response of a Differentiator

Response of Deterministic LTI systems to white noise-example

Response of Deterministic LTI systems to white noise-example

Response of Deterministic LTI systems to white noise

Response of Deterministic LTI systems to white noise

Deterministic Linear Time Invariant Systems with Stochastic Inputs

Deterministic Linear Time Invariant Systems with Stochastic Inputs

Linear Systems with Stochastic Inputs

Linear Systems with Stochastic Inputs

Bussgang's Theorem-two examples

Bussgang's Theorem-two examples

Bussgang's theorem

Bussgang's theorem

Hard Limiter

Hard Limiter

Stationarity in memoryless time invariant systems

Stationarity in memoryless time invariant systems

Square-law detector

Square-law detector

Memoryless Systems

Memoryless Systems

Systems with Stochastic Inputs

Systems with Stochastic Inputs

More Definitions for Stochastic Processes

More Definitions for Stochastic Processes

A narrowband Stationary Signal

A narrowband Stationary Signal

Integration of a WSS RP

Integration of a WSS RP

Jointly Wide Sense Stationary Processes

Jointly Wide Sense Stationary Processes

Wide Sense Stationary Process

Wide Sense Stationary Process

Stationary process

Stationary process

Point and renewal process

Point and renewal process

Example of a Normal Random Process

Example of a Normal Random Process

Normal Random Process

Normal Random Process

Uncorrelated increment process

Uncorrelated increment process

Integration of white noise

Integration of white noise

Stochastic processes more nomenclature

Stochastic processes more nomenclature

Integration

Integration

Stochastic processes extended nomenclature

Stochastic processes extended nomenclature

Nonuniform Poisson Point Process and Telegraph Signal

Nonuniform Poisson Point Process and Telegraph Signal

Poisson Point Process

Poisson Point Process

Independent increment process

Independent increment process

Second-order nomenclature

Second-order nomenclature

Statistics of stochastic processes

Statistics of stochastic processes

Stochastic Processes Nomenclature

Stochastic Processes Nomenclature

Stochastic Processes

Stochastic Processes

Innovation Process

Innovation Process

Gram Schmidt Orthogonalization

Gram Schmidt Orthogonalization

Linear Data Transformation

Linear Data Transformation

Orthogonality Principal for a Non Linear Estimator

Orthogonality Principal for a Non Linear Estimator

Nonlinear Mean Square Estimator

Nonlinear Mean Square Estimator

Conditional Densities of Normal Random Vectors

Conditional Densities of Normal Random Vectors

Orthogonality Principal for a Linear Estimator

Orthogonality Principal for a Linear Estimator

Linear Mean Square Estimation

Linear Mean Square Estimation

Sample Mean and Variance of iid RVs

Sample Mean and Variance of iid RVs

Circularly Symmetrical Jointly Normal Distribution

Circularly Symmetrical Jointly Normal Distribution

Jointly Normal Distribution

Jointly Normal Distribution

Characteristic Functions

Characteristic Functions

Conditional Densities and Moments

Conditional Densities and Moments

Conditional Densities

Conditional Densities

Properties of Correlation and Covariance Matrices

Properties of Correlation and Covariance Matrices

Correlation and Covariance Matrices

Correlation and Covariance Matrices

Independent Versus Uncorrelated

Independent Versus Uncorrelated

Covariance Matrix Sample Mean and Variance

Covariance Matrix Sample Mean and Variance

Complex Valued Random Variables and Vectors

Complex Valued Random Variables and Vectors

Minimum Variance Linear Unbiased Estimator

Minimum Variance Linear Unbiased Estimator

Ordered Statistics

Ordered Statistics

Independent Random Variables

Independent Random Variables

Transformation of Random Vectors

Transformation of Random Vectors

Random Vectors

Random Vectors

Generalization of central limit theorem and product of iid RVs

Generalization of central limit theorem and product of iid RVs

A Limit Theorem for Product of Random Variables

A Limit Theorem for Product of Random Variables

Central Limit Theorem (CLT) & Proof of CLT

Central Limit Theorem (CLT) & Proof of CLT

Ergodicity

Ergodicity

Law of Large Numbers

Law of Large Numbers

Suﬃcient Condition for Convergence Almost Everywhere

Suﬃcient Condition for Convergence Almost Everywhere

Necessary and Suﬃcient Condition for Convergence in Probability

Necessary and Suﬃcient Condition for Convergence in Probability

Convergence of Sums of Orthogonal RVs

Convergence of Sums of Orthogonal RVs

Necessary and Suﬃcient Condition for MS Convergence

Necessary and Suﬃcient Condition for MS Convergence

Example

Example

Relationships Between Modes of Convergence

Relationships Between Modes of Convergence

Modes of Stochastic Convergence

Modes of Stochastic Convergence

Example of Stochastic Convergence

Example of Stochastic Convergence

Stochastic Convergence and Limit Theorems

Stochastic Convergence and Limit Theorems

Random Vectors Learning outcomes

Random Vectors Learning outcomes

Summary

Summary

Risk and loss functions

Risk and loss functions

Linear Minimum Mean Square Estimation

Linear Minimum Mean Square Estimation

Nonlinear Minimum Mean Square Estimation

Nonlinear Minimum Mean Square Estimation

Minimum Mean Square Error Estimation

Minimum Mean Square Error Estimation

Reliability of interconnected systems

Reliability of interconnected systems

Example for Hazard Rate

Example for Hazard Rate

Hazard Rate of a System

Hazard Rate of a System

Example for Reliability

Example for Reliability

Reliability Engineering

Reliability Engineering

Mean of conditional mean

Mean of conditional mean

Forming RVs using conditional expected value

Forming RVs using conditional expected value

Conditional expected value and moments

Conditional expected value and moments

Baye's theorem and total probability

Baye's theorem and total probability

Example for conditional probability density function

Example for conditional probability density function

Conditional probability density function

Conditional probability density function

Examples for Joint characteristic functions

Examples for Joint characteristic functions

Joint characteristic functions - Part II

Joint characteristic functions - Part II

Variance of sum of RVs

Variance of sum of RVs

Introduction to Conditional distributions

Introduction to Conditional distributions

Joint moment generating functions

Joint moment generating functions

Joint characteristic functions - Part I

Joint characteristic functions - Part I

Estimating moments using other moments

Estimating moments using other moments

Independence and joint moments

Independence and joint moments

Covariance, correlation and independence

Covariance, correlation and independence

Space of Random Variables and Inner Product

Space of Random Variables and Inner Product

Joint moments and central moments

Joint moments and central moments

Definition of Joint Moments

Definition of Joint Moments

Overview and Chapter Learning Outcomes

Overview and Chapter Learning Outcomes

Transformation of Random Pair between Cartesian and Polar

Transformation of Random Pair between Cartesian and Polar

Definition of Expected Value Operator for discrete type distributions

Definition of Expected Value Operator for discrete type distributions

Summary of Two Random Variables and Random Vectors

Summary of Two Random Variables and Random Vectors

Using auxiliary random variables

Using auxiliary random variables

Examples - transformation of two random variables - part 1

Examples - transformation of two random variables - part 1

Linear transformation

Linear transformation

Other Conditional distributions

Other Conditional distributions

Step by step joint density of two functions of two random variables

Step by step joint density of two functions of two random variables

Joint density of two functions of two random variables

Joint density of two functions of two random variables

Example - Two functions of two random variables

Example - Two functions of two random variables

Distribution of maximum of random variables

Distribution of maximum of random variables

Example - Narrowband zero mean Gaussian Noise

Example - Narrowband zero mean Gaussian Noise

Example - Transformation of Circularly Symmetrical Random Pair between Cartesian and Polar

Example - Transformation of Circularly Symmetrical Random Pair between Cartesian and Polar

Cartesian and Polar Coordinates

Cartesian and Polar Coordinates

Sum of two independent exponentially distributed random variables

Sum of two independent exponentially distributed random variables

DAD and sum of two random variables

DAD and sum of two random variables

The distribution of the Euclidian norm for non circularly symmetrical normal bivariate

The distribution of the Euclidian norm for non circularly symmetrical normal bivariate

The pdf of division of two jointly normal random variables

The pdf of division of two jointly normal random variables

The distribution of the Euclidian norm

The distribution of the Euclidian norm

The pdf of division of two circularly symmetrical random variable

The pdf of division of two circularly symmetrical random variable

The pdf of division of two random variables

The pdf of division of two random variables

Sum of two independent uniformly distributed random variables

Sum of two independent uniformly distributed random variables

One function of two random variables

One function of two random variables

Independence of two random variables

Independence of two random variables

Joint PDF

Joint PDF

Independence of two separate functions of two independent random variables

Independence of two separate functions of two independent random variables

Circular Symmetry and Independence of random variables

Circular Symmetry and Independence of random variables

Properties of Joint CDF

Properties of Joint CDF

Bivariate CDF

Bivariate CDF

Introduction to Random Vectors

Introduction to Random Vectors

Generation of RVs and Summary

Generation of RVs and Summary

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

Examples of finding the pdf of a function of a RV

A presedure to find the pdf of a function of a RV

A presedure to find the pdf of a function of a RV

Distribution of absolute value of x where x is normal

Distribution of absolute value of x where x is normal

Distribution of squared of x where x is uniformly distributed

Distribution of squared of x where x is uniformly distributed

Distribution of squared of x

Distribution of squared of x

Distribution of a linear transformation

Distribution of a linear transformation

Cdf of a transformed RV

Cdf of a transformed RV

Definition of a transformed RV

Definition of a transformed RV

Type of a transformed RV

Type of a transformed RV

Introduction of Chapter 4

Introduction of Chapter 4

Conditional distributions

Conditional distributions

Existence theorem

Existence theorem

Pdf of discrete and mixed type RVs

Pdf of discrete and mixed type RVs

Types of a RV

Types of a RV