Distributions of Linear Transformations of Random Variables

In this lecture we look at how to identify the distributions of linear transformations and linear combinations of Normal random variables.

Here is an outline of what we will be covering in this lecture:

The Use of Linear Transformations of Normal Random Variables
Identify the Distribution of:
Multiple Linear Transformations of General Multivariate Normal Random Variables - Theorem 1.4
A Single Linear Transformation of General Multivariate Normal Random Variables - Corollary 1.4.1
Multiple Linear Transformations of Uncorrelated Normal Random Variables with Equal Variances - Corollary 1.4.2
A Single Linear Transformation of Uncorrelated Normal Random Variables with Equal Variances - Corollary 1.4.3

0:00 Introduction
0:10 Outline
1:13 Use of Linear Transformation of Normal Random Variables
4:11 Distribution of Multiple Linear Transformations of General Multivariate Normal Random Variables
16:56 Distribution of a Single Linear Transformation of General Multivariate Normal Random Variables
23:36 Distribution of Multiple Linear Transformations of Uncorrelated Normal Random Variables with Equal Variables
30:48 Distribution of a Single Linear Transformation of Uncorrelated Normal Random Variables with Equal Variables
35:45 Important Takeaway
36:01 Relationships Between the Results
38:07 Outro