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arithmetic mean/main.tex

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+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+% Core packages
+\usepackage{amsmath,amssymb}
+\usepackage{tikz-cd}
+\usepackage{multicol}
+
+% Paragraphs
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{1\baselineskip}
+
+\title{Arithmetic Mean}
+\author{algorembrant}
+\date{\today}
+
+\begin{document}
+\maketitle
+
+
+
+
+
+
+
+The average of total lookback $T$ is defined as:
+\[
+\text{average} = \frac{1}{T} \sum_{t=0}^{T-1} x_{0-t}, \quad \text{where $t$ is $timestamp$ $indexing$ and $0$ as current time}
+\]
+which expands as
+\[
+    \frac{x_0 + x_{-1} + x_{-2} + x_{-3} + \ldots + x_{-(T-1)}}{T}, \quad \text{like} \quad
+    \frac{x_0 + x_{-1} + x_{-2} + x_{-3}}{4}
+\]
+and other variants
+
+\[
+\text{average} = 
+\begin{cases}
+    \text{if lookback}, & \displaystyle \frac{1}{T} \sum_{t=0}^{T-1} x_{0-t} = \frac{x_0 + x_{-1} + x_{-2} + \ldots + x_{-(T-1)}}{T} \\[10pt]
+    \text{if lookahead}, & \displaystyle \frac{1}{T} \sum_{t=0}^{T-1} x_{0+t} = \frac{x_0 + x_{1} + x_{2} + x_{3} + \ldots + x_{T-1}}{T}
+\end{cases}
+\]
+or starts indexing at 1
+\[
+\text{average} = 
+\begin{cases}
+    \text{if lookback}, & \displaystyle \frac{1}{T} \sum_{t=1}^{T} x_{-t} = \frac{x_{-1} + x_{-2} + x_{-3} + \ldots + x_{-(T)}}{T} \\[10pt]
+    \text{if lookahead}, & \displaystyle \frac{1}{T} \sum_{t=1}^{T} x_{t} = \frac{x_{1} + x_{2} + x_{3} + \ldots + x_{T}}{T}
+\end{cases}
+\]
+\end{document}