From 67faf390ff05f25db809e9faad5c21eb365d76bf Mon Sep 17 00:00:00 2001
From: andyeisenberg <125768484+andyeisenberg@users.noreply.github.com>
Date: Thu, 1 Jan 2026 05:00:56 +0000
Subject: [PATCH] Starting with variance
---
source/ch-Expected-Value.ptx | 48 +++++++++++++++++++++++++++++++++++-
1 file changed, 47 insertions(+), 1 deletion(-)
diff --git a/source/ch-Expected-Value.ptx b/source/ch-Expected-Value.ptx
index ae04669..38170fe 100644
--- a/source/ch-Expected-Value.ptx
+++ b/source/ch-Expected-Value.ptx
@@ -538,9 +538,55 @@
Variance
- Text of section.
+ The expected value is one of several statistics sometimes referred to as a "measure of central tendency".
+ These are statistics that try to capture the "typical" value of the random variable.
+ Our next statistic tries to measure how spread out a distribution is.
+
+
+
+ Let X be a random variable.
+ The variance of X is:
+
+ \Var(X) = \E\left[ \left(X - \E(X)\right)^2 \right].
+
+
+
+
+
+
+ This formula tries to capture some measure of how spread out the distribution is.
+ One way we might hope to measure spread is to calculate how far away X is from its expected value on average.
+ Squaring the term X - \E(X) ensures that any difference between the value of the random variable and its expected value contributes positively.
+ ("Distance away from" should be positive.) So, the formula in is conceptually clear.
+ In practice, there's another way to structure the formula that makes for cleaner calculation:
+
+
+
+
+
+ Let X be a random variable.
+ Then:
+
+ \Var(X) = \E\left(X^2\right) - \left(\E(X)\right)^2.
+
+
+
+
+
+
+
+
+
+ \Var(X) \amp = \E\left[ \left(X - \E(X)\right)^2 \right]
+ \amp = \E\left[ X^2 - 2 \E(X) X + \left(\E(X)\right)^2\right]
+ \amp = \E\left(X^2\right) - 2 \E(X) \E(X) + \left(\E(X)\right)^2
+ \amp = \E\left(X^2\right) - \left(\E(X)\right)^2.
+
+
+
+