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 0000000..62f0cb8 Binary files /dev/null and b/report/report.pdf differ 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&.5