From 2f9d5e00b0393d536063c6e0c4642b38f3a78130 Mon Sep 17 00:00:00 2001
From: Kmart <blulightspecial@users.noreply.github.com>
Date: Thu, 11 Jan 2024 20:13:18 -0800
Subject: [PATCH] Small Latex Update to problem 3-G-1 Definition of Cov

---
 .../unit-03/G-definition-of-covariance/1.md   | 40 +++++++++----------
 1 file changed, 20 insertions(+), 20 deletions(-)

diff --git a/_problems/unit-03/G-definition-of-covariance/1.md b/_problems/unit-03/G-definition-of-covariance/1.md
index c3dc519..61f614b 100644
--- a/_problems/unit-03/G-definition-of-covariance/1.md
+++ b/_problems/unit-03/G-definition-of-covariance/1.md
@@ -14,26 +14,26 @@ Part a
 
 \\[
 \begin{align}
-\text{Thrm 2.2.2}\\
-Cov[X,Y] &= E[XY] - E[X]E[Y] \\
-&=E[X(X^2)] - E[X]E[X^2]\\
-\\
-\text{Thrm 2.1.1}\\
-E[X] &= \sum_x xf(x)\\
-     &= (-2)*0.2 + (-1)*0.2 + 0*0.2 + 1*0.2 + 2*0.2 \\
-     &= 0\\
-\\
-\text{Thrm 2.1.5} & \; \text{(LOTUS)} \\
-E[X^2] &= \sum_x g(x)f(x)\\
-       &= 4*0.2 + 1*0.2 + 0*0.2 + 1*0.2 + 4*0.2 \\
-       &= 2\\
-\\
-E[X^3] &= \sum_x g(x)f(x)\\
-       &= (-8)*0.2 + (-1)*0.2 + 0*0.2 + 1*0.2 + 8*0.2 \\
-       &= 0\\
-\\
-\text{Plug in}\\
-Cov[X,Y] &= 0 - 0*2\\
+\text{Thrm 2.2.2}\\\\\
+Cov[X,Y] &= E[XY] - E[X]E[Y] \\\\\
+&=E[X(X^2)] - E[X]E[X^2]\\\\\
+\\\\\
+\text{Thrm 2.1.1}\\\\\
+E[X] &= \sum_x xf(x)\\\\\
+     &= (-2)*0.2 + (-1)*0.2 + 0*0.2 + 1*0.2 + 2*0.2 \\\\\
+     &= 0\\\\\
+\\\\\
+\text{Thrm 2.1.5} & \; \text{(LOTUS)} \\\\\
+E[X^2] &= \sum_x g(x)f(x)\\\\\
+       &= 4*0.2 + 1*0.2 + 0*0.2 + 1*0.2 + 4*0.2 \\\\\
+       &= 2\\\\\
+\\\\\
+E[X^3] &= \sum_x g(x)f(x)\\\\\
+       &= (-8)*0.2 + (-1)*0.2 + 0*0.2 + 1*0.2 + 8*0.2 \\\\\
+       &= 0\\\\\
+\\\\\
+\text{Plug in}\\\\\
+Cov[X,Y] &= 0 - 0*2\\\\\
          &= 0
 \end{align}
 \\]