From ead29fd6379edd9411437da234ef75cf5b50c908 Mon Sep 17 00:00:00 2001 From: Andrew Simonson Date: Thu, 3 Oct 2024 23:47:33 -0400 Subject: [PATCH] unit 1 report --- notes.md | 9 +++ report/report.pdf | Bin 0 -> 119460 bytes report/report.tex | 149 ++++++++++++++++++++++++++++++++++++++++ timesheet/timesheet.csv | 10 ++- timesheet/timesheet.pdf | Bin 56753 -> 44264 bytes timesheet/timesheet.tex | 28 ++++++-- 6 files changed, 187 insertions(+), 9 deletions(-) create mode 100644 notes.md create mode 100644 report/report.pdf create mode 100644 report/report.tex diff --git a/notes.md b/notes.md new file mode 100644 index 0000000..19e80cd --- /dev/null +++ b/notes.md @@ -0,0 +1,9 @@ +## Assumptions and Learnings + - probability density function expected value + - Confidence interval is not my value * confidence, it's confidence chance of being in my range + - I've made some mistakes in stat review, looking at narrow topics before covering broader parent topics. Should reorganize structure (tree, not list?) + +t-test, z-test: both are hypothesis tests +The t-test is used when the population variance is unknown, or the sample size is small (n < 30) +The z-test is used when the population variance (σ2) is known *and* the sample size is large (n > 30) +To create a z-distribution table, mathematicians calculate the CDF for various z-scores and tabulate the results. \ No newline at end of file diff --git a/report/report.pdf b/report/report.pdf new file mode 100644 index 0000000000000000000000000000000000000000..62f0cb8dd05b8e19489f681c08598154e2f9ef2a GIT binary patch literal 119460 zcma(1Q_wI>tThTgwr$(CZQHhO+qP}veQevdZR>n9bI#oCU3LCkbyqG@wK~a4(jiq4 z5u;_KV}&A}UmjV9V&NoUAh0*Gg5u$UqL(qXGk38dU}EPa`2PtMy_ltqi>VUZjIvAu~Y6dxayvx}3dp)HihrmTXr{15|7@0t30h6Ky7XY0u*f`r3C<1QK8R#v2x zt&$YNX5W}?c!*QP%f{Z&F3a`j0D%D|D;M+(eO#wlZi>j&A z+eHR9Uki1iDE6X;(T6i*x$qG#8RzQbQ!CsBsja@JZ=j+_!(}K_JCpw#q5n?*A9UIN z7noT%nE#JqUP+s>LuNz?d-H^%zX^?vw>NG+T9hInsWO`c)8L4N4o=FXz?Aazo{iZ_ 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ze6klf`9r?k_$SzY{KC>n(>F$$^BfYM`e)A7ss-|sHt?Un$!-4bO;`K@Z;4jLsTX#f ztz}*RkiqV-bOra}4{~-Jrr5qP=~?O0yzPL6k3p8dHS=Zz_Rkv_x;L<#-oUh4YvYu) zo0e_*`2WIdE%l|-%tDM3y=QCfPo8GCYMOk+tsvjJja$}Q&EK+Ran+QqUp&8NAG)+I z%j#Y9m)m{Mt8W#5J3QI;%b%amciMc7xm!{9Yxea!@#oiPReM`qyeQVUt-z#tv82q| x2N{g++&ikSzAmua0PABeLqiL5OW=WVs;>TSTma@5RwV!c literal 0 HcmV?d00001 diff --git a/report/report.tex b/report/report.tex new file mode 100644 index 0000000..438381d --- /dev/null +++ b/report/report.tex @@ -0,0 +1,149 @@ +\documentclass[12pt]{article} +\usepackage{blindtext} +\usepackage{hyperref} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage[a4paper, total={6in, 10in}]{geometry} +\hyphenpenalty 1000 + +\begin{document} +\begin{titlepage} +\begin{center} + +\vspace*{5cm} +\Large{\textbf{Implementations of Probability Theory}}\\ + +\rule{14cm}{0.05cm}\\ \vspace{.25cm} + +\Large{Independent Study Report}\\ +\large{Andrew Simonson} + +\vspace*{\fill} +\large{Compiled on: \today}\\ + +\end{center} +\end{titlepage} + +\newpage +% Table of Contents +% \large{Table of Contents} +\tableofcontents +\addtocontents{toc}{~\hfill\textbf{Page}\par} + +\newpage +% Begin report +\section{Objective} +yada yada yah I started this independent study for my own selfish gain + +\newpage +\section{Units} +\rule{14cm}{0.05cm} +\subsection{Unit 1: Statistics Review} +To ensure a strong statistical foundation for the future learnings in probabilistic models, +the first objective was to create a document outlining and defining key topics that are +prerequisites for probabilities in statistics or for understanding generic analytical models. + +\subsubsection{Random Variables} +\begin{enumerate} +\item \textbf{Discrete Random Variables - }values are selected by chance from a countable (including countably infinite) list of distinct values +\item \textbf{Continuous Random Variables - }values are selected by chance with an uncountable number of values within its range +\end{enumerate} + +\subsubsection{Sample Space} +A sample space is the set of all possible outcomes of an instance. For a six-sided dice roll event, +the die may land with 1 through 6 dots facing upwards, hence: +\[S = [1, 2, 3, 4, 5, 6] \quad\text{where }S\text{ is the sample space}\] + +\subsubsection{Probability Axioms} +There are three probability axioms: + +\begin{enumerate} + \item \textbf{Non-negativity}: + \[ + P(A) \geq 0 \quad \text{for any event }A, \ P(A) \in \mathbb{R} + \] + No event can be less likely to occur than an impossible event ( \(P(A) = 0\) ). P(A) is a real number. + Paired with axiom 2 we can also conclude that \(P(A) \leq 1\). + + \item \textbf{Normalization}: + \[ + P(S) = 1\quad\text{where }S\text{ is the sample space} + \] + \textbf{Unit Measure - } All event probabilities in a sample space add up to 1. In essence, there is a 100\% + chance that one of the events in the sample space will occur. + + \item \textbf{Additivity}: + \[ + P(A \cup B) = P(A) + P(B) \quad \text{if } A \cap B = \emptyset + \] + A union between events that are mutually exclusive (events that cannot both happen for an instance) has a + probability that is the sum of the associated event probabilities. +\end{enumerate} + +\subsubsection{Expectations and Deviation} +\begin{enumerate} +\item \textbf{Expectation - }The weighted average of the probabilities in the sample space +\[\sum_{}^{S}{P(A) * A} = E \quad\text{where }E\text{ is the expected value}\] +\item \textbf{Variance - }The spread of possible values for a random variable +\item \textbf{Standard Deviation - }something +\[std = \sqrt{V}\quad\text{where variance is }V\] +\end{enumerate} + +\subsubsection{Probability Functions} +Probability Functions map the likelihood of random variables to be a specific value. + +\subsubsection*{Probability Mass Functions} +Probability Mass Functions (PMFs) map discrete random variables. +For example, a six-sided die roll creates a uniform random PMF: +\begin{equation*} + P(A) = + \begin{cases} + 1/6\qquad\text{if }&X=1\\ + 1/6&X=2\\ + 1/6&X=3\\ + 1/6&X=4\\ + 1/6&X=5\\ + 1/6&X=6\\ + \end{cases} +\end{equation*} + +\subsubsection*{Probability Density Functions} +Probability Density Functions (PDFs) map continuous random variables. +For example, this is a PDF where things happen. +\begin{equation*} + P(A) = + \begin{cases} + X\qquad\qquad\text{if }&0\leq X\leq .5\\ + -X+1&.5lew5@l1aW7 zGoOYo@UsoDcrs9qBu7wcfNI=;{4gr}I>X4C5f z!(F}3m7x9pBcg|IqON{BA{fw|WebF0!g$4t(tu7AHE(DFpS!Yiwo&TY!1z;N!bKlU z4DKnFzMVQ5tJiBLM9uhG9v9k@lE#2>BK|zJZ3dri`ky~~9n**u8ohJ>lO;n-`u+opOUpqA%u%6grAwH<% zhws?NdOff@;J8wytLp8Ea@YAZzPT1byZ^Nl-9Vc>DEF9C4zCGx8n2+wi5)Z5<(B8% zLaS+h|9nD{(orj=fi<-=advSsHMISo$lk~bmYJEDiTHm4K0a7RSyMZ67Ykx$HWs!d z`$Z~1O{TW}76-cj?j!9J6)5`?WsSSwQH44H^;6S$9&XCREofrmxCoN{ObLtI8X_?h#;bsj4glK 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a/timesheet/timesheet.tex b/timesheet/timesheet.tex index 080191d..112c708 100644 --- a/timesheet/timesheet.tex +++ b/timesheet/timesheet.tex @@ -1,13 +1,15 @@ -\documentclass{article} +\documentclass[12pt]{article} \usepackage{blindtext} -\usepackage[a4paper, total={6in, 8in}]{geometry} +\usepackage[a4paper, total={6in, 10in}]{geometry} \nofiles +\hyphenpenalty 1000 \begin{document} \begin{titlepage} \begin{center} +\vspace*{5cm} \Large{\textbf{Implementations of Probability Theory}}\\ \rule{14cm}{0.05cm}\\ \vspace{.5cm} @@ -28,18 +30,30 @@ \hline Week & Date & Type & Duration (Hours) & Description \\ \hline -1 & 08/30 & Advising Meetings & 1 & Stat Review Content acknowledgement, Latex overview for reports \\ +1 & 08/30 & Advising Meetings & 2 & Stat Review Content acknowledgement, Latex overview for reports \\ \hline 2 & 09/02 & Reporting & 3 & First applications of Latex for final report, created Timesheet System. \\ \hline 2 & 09/02 & Research & 2 & Stat Review: Sample Space through Probability Density Functions \\ \hline +2 & 09/06 & Advising Meetings & 1 & Research Review and exploration of PDF expected values and confidence intervals \\ +\hline +4 & 09/19 & Research & 2 & Producing Confidence Intervals \\ +\hline +4 & 09/20 & Research & 1 & Statistical Inference and t-testing \\ +\hline +4 & 09/20 & Advising Meetings & 1 & Stat Review finalization, definition of reporting standard \\ +\hline +5 & 09/23 & Research & 2 & Parametric and Non-parametric tests \\ +\hline +6 & 10/03 & Reporting & 4 & Structuring stat review report \\ +\hline \end{tabular} \end{table} -\noindent Hours for Advising Meetings: 1\\ -Hours for Reporting: 3\\ -Hours for Research: 2\\ -\textbf{Total Hours: 6}\\ +\noindent Hours for Advising Meetings: 4\\ +Hours for Reporting: 7\\ +Hours for Research: 7\\ +\textbf{Total Hours: 18}\\ % CLOSE Timesheet \end{document} \ No newline at end of file