Reading from text, Lecture 4

Chapter 4

This chapter is about the Algebra of Expectation.

4.1

Definition 4.1 (Defines the mean of a random variable)

Theorem 4.1 (extends the definition to include "transformations" of the random variable, where the x-values change, but the probabilities do not)

Example 4.4

Example 4.5

Definition 4.2 (Notice that although f is a joint random variable, g is not - it takes on scalar values, it is "simple". That is, this is not the mean of a joint random variable! It is the mean of a "transformation" of the joint random variable that creates a univariate RV.)

Example 4.6

4.2

Introduction

Definition 4.3 (definition of variance of a RV, and standard deviation)

Example 4.8

Theorem 4.2 (much simpler definition to work with)

Example 4.9, 4.10

Theorem 4.3 (extends the definition of variance to include "transformations" of the RV where the x-values change but the probabilities do not)

Example 4.11

Definition 4.4 (definition of covariance)

Theorem 4.4 (again, much simpler definition to work with)

Example 4.13

Definition 4.5 (definition of correlation)

Example 4.15

4.3

Theorem 4.5 (and what follows ... for means, everything is linear as expected)

Corollary 4.1, 4.2

Example 4.17, 4.18

Theorem 4.6 (expected value for sums and differences of functions)

Example 4.19

Theorem 4.7 (extends 4.6 to joint functions)

Corollary 4.3, 4.3

Theorem 4.8 (still working with means, but not linear combos: XY instead. Independence or lack thereof becomes important.)

Corollary 4.5

Theorem 4.9 (we are back to linear combos, but now for variances. Not so nice as means.)

Corollary 4.6, 4.7, 4.8,

Corollary 4.9, 4.10, 4.11 (special case, Independence, where simplifications occur)

Example 4.22

Example 4.23

4.4

(all)