Learn · Probability Theory for Data Science
Lecture 49 of 63Week 10
Moments of a Multivariate Random Variable
Prof. Ishapathik Das · IIT Tirupati
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- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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All the lectures
- Introduction
- Sample Space and Events
- Special Events and Various Approaches to Defining Probability
- Important Theorems
- Numerical Examples and Introduction to Conditional Probability
- Definition of Conditional Probability and Independence
- Bayes' Theorem
- Random Variable
- Events Defined by a Random Variable
- Cumulative Distribution Function
- Properties of the Cumulative Distribution Function and Discrete Random Variable
- Probability Mass Function
- Continuous Random Variable and Probability Density Function
- Numerical Examples
- Moments
- Higher Order Moments and Variance of a Random Variable
- Numerical Examples of Moments and Bernoulli Distribution
- Binomial Distribution
- Applications of Binomial Distribution
- Poisson Distribution
- Applications of Poisson Distribution
- Numerical Examples of Poisson Distribution and Uniform Distribution
- Application of Uniform Distribution and Exponential Distribution
- Applications of Exponential Distribution
- Memoryless Property and Gamma Distributio
- Example of Gamma Distribution and Normal Distribution
- Properties of Normal Distributions
- Numerical Examples of Normal Distributions
- Applications of Normal Distributions and Conditional Distribution Function
- Examples of Conditional Distribution Function and Bivariate Random Variable
- Example of Bivariate Random Variable
- Properties of the Joint Cumulative Distribution Function of a Bivariate Random Variable
- Independence Between Two Random Variables
- Examples of Joint Cumulative Distribution Functions, Marginals, and Independence
- Joint Probability Mass Function, Marginal Probability Mass Function, Examples
- Numerical Examples on Bivariate Discrete Random Variables and the Concept of Joint Probability Density Function
- Marginal Probability Density Function, Independence, and Examples
- Numerical Examples on Probability Density Function
- Conditional Probability Mass Function
- Conditional Probability Density Function
- Moments for Bivariate Random Variables
- Association Between Two Random Variables
- Numerical Examples on Moments for Bivariate Random Variables
- Conditional Mean and Variance for Discrete Random Variables
- Conditional Mean and Variance for Continuous Random Variables
- Numerical Examples on Conditional Mean and Variance
- Multivariate Random Variables
- Multivariate Probability Density Function and Independence
- Moments of a Multivariate Random Variable
- Numerical Examples on Joint Probability Mass Functions
- Numerical Examples on Joint Probability Density Functions
- Multinomial Distribution and Multivariate Normal Distribution
- Transformation of Random Variables
- Theorem on Transformation of Random Variables
- Transformation of Multivariate Random Variables
- Examples of Transformation of Bivariate Random Variables
- Convolution and Example on Transformation of n-variate Random Variables
- Transformation of Discrete Random Variables
- Moment Generating Functions
- Example of Moment Generating Functions
- Moment Generating Functions for the Transformation of Random Variables
- Chebyshev's Inequality
- Notions of Convergence, Law of Large Numbers, and the Central Limit Theorem