From ccd737b4aac6d4ac3a0775b6aacd60bd6fd38c86 Mon Sep 17 00:00:00 2001
From: eggy <danielchen04@hotmail.ca>
Date: Mon, 10 Apr 2023 13:30:32 -0400
Subject: [PATCH] ece108: add expected value

---
 docs/1b/ece108.md | 24 ++++++++++++++++++++++++
 1 file changed, 24 insertions(+)

diff --git a/docs/1b/ece108.md b/docs/1b/ece108.md
index 76467e2..85fb09e 100644
--- a/docs/1b/ece108.md
+++ b/docs/1b/ece108.md
@@ -932,3 +932,27 @@ Formally, this can be solved without $Pr\{B\}$:
 
 $$Pr\{A|B\}=\frac{Pr\{A\}Pr\{B|A\}}{Pr\{A\}Pr\{B|A\}+Pr\{\overline A\}Pr\{B|\overline A\}}$$
 
+### Expected value
+
+The **expected value**, **mean**, or **expectation of $X$** is:
+
+$$E[X]=\sum_{x\in\mathbb R}x\cdot Pr\{X=x\}=\sum_{s\in S}X(s)\cdot Pr\{\{s\}\}$$
+
+This operation is **linear**, but multiplies using AND:
+
+$$
+E[X+Y]=E[X}+E[Y] \\
+E[XY]=\sum_{x\in X,y\in Y}xy\cdotPr\{X=x\wedge y\=y\}
+$$
+
+Thus if $X$ and $Y$ are independent:
+
+$$E[XY]=E[X]E[Y]$$
+
+An **indicator random variable** only has two possible outcomes: zero or one. Thus an indicator random variable $X$ has an expected value equal to its probability:
+
+$$E[X]=Pr\{X=1\}$$
+
+The **covariance** of $X$ and $Y$ represents the direction of difference of $X$ and $Y$ from their means.
+
+$$Cov[X,Y]=E[XY]-E[X]E[Y]$$