## The Derivation of the Normal Distribution

\documentclass{article}\usepackage[utf8]{inputenc}

% !BIB TS-program = biber

\usepackage[backend=biber,style=numeric, citestyle=authoryear]{biblatex}

\addbibresource{blog.bib}

\usepackage{amssymb}

\usepackage{dirtytalk}

\usepackage{csquotes}

\usepackage{amsmath}

\usepackage{calc}

\usepackage{textcomp}

\usepackage{mathtools}

\usepackage[english]{babel}

\usepackage{fancyhdr}

\usepackage{url}

\def\UrlBreaks{\do\/\do-}

\usepackage{breakurl}

\usepackage[breaklinks]{hyperref}

\usepackage{graphicx}

\graphicspath{ {images/} }

\usepackage{wrapfig}

\usepackage{float}

\usepackage[T1]{fontenc}

\usepackage{outlines}

\usepackage{enumitem}

\setenumerate[1]{label=\null}

\setenumerate[2]{label=\null}

\setenumerate[3]{label=\roman*.}

\setenumerate[4]{label=\alph*.}

\newcommand{\midtilde}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}

\pagestyle{fancy}

\fancyhf{}

\title{Statistical Analysis by Communicative Functionals: \\ Lecture 1 – A Geometric Derivation of the Normal Distribution}

\author{Justin Petrillo}

\begin{document}

\maketitle

\begin{abstract}

The Internal Space-Time Geometry to Experiment as the Distribution of Measurement InterActions is set up by the Statistical Parameter.

\end{abstract}

\section{The Scientific Process}

Statistics is the method of determining the validity of an empirical claim about nature. A claim that is not particularly valid will likely be true only some of the time or under certain specific conditions that are not too common. Ultimately, thus, within a domain of consideration, statistics answers the question of the universality of the claims made about nature through empirical methods of observation. It may be that two opposing claims are both true in the sense that they are each true half the time of random observation or within half the space of contextual conditionalities. The scientific process, as progress, relies on methods that over a linear time of repeated experimental cycles, increase the validity of the claims as the knowledge of nature approaches universality, itself always merely a horizon within the phenomenology of empiricism. This progressive scientific process is called ‘discovery,’ or merely $research$, although it is highly non-linear.

The scientific process is a branching process as the truth of a claim is found to be dependent upon its conditions, and those conditions found dependent on further conditionals. This structure of rationality is as a tree. A single claim (C) has a relative validity (V) due to the truth of an underlying, or conditioning, claim, $C_i$, given as $V_{C_i}(C)=V(C,C_i)$. We may understand the validity of claims through probability theory, in that the relative validity of a claim based on a conditioning claim is the probability the claim is true conditioned on $C_i$, $V(C,C_i)=P(C|C_i)$. In general, we will refer to the object under investigation, of which C is a claim about, as the primary variable X, and the subject performing the investigation, of which $C_i$ is hypothesized (as a cognitive action), as the secondary variable Y. Thus, the orientation of observation, i.e. the time-arrow, is given as $\sigma: Y \rightarrow X$.

An observer (Y) makes an observation from a particular position of an event (X) with its own place, forming a space-time of the action of measurement. An observation-as-information is a complex quantum-bit, which within a space of investigation is a complex variable, representing a tree of observation-conditioning rationality resulting from the branching process of hypothesis formation, with each node a conditional hypothesis and edge length the conditional probability. The gravitation of the system of measurement is the space-time tensor of its world-manifold, stable or chaotic of the time of interaction. We thus understand the positions of observers within a place of investigation, itself given at least in real-part component by the object of investigation.

\section{Experimental Set-up}

Nature is explained by a parameterized model. Each parameter, as a functional aggregation of measurement samples, has itself a corresponding distribution as it $occurs \ in \ nature$ along the infinite, universal horizon of measurement. \\

\\

Let $X^n$ be a random variable representing the n qualities that can be measured for the thing under investigation, $\Omega$, itself the collected gathering of all its possible appearances, $\omega \in \Omega$ such that $X^n:\omega \rightarrow {\mathbb{R}}^n$. Each sampled measurement of $X^n$ through an interaction with $\omega$ is given as an $\hat{X}^n(t_i)$, each one constituting a unit of indexable time in the catalogable measurement process. Thus, the set of sampled measurements, a $sample \ space$, is a partition of ‘internally orderable’ test times within the measurement action, $\{ \hat{X}^n(t): t \in \pi \}$. \\

\\

In this set up of statistical sampling, one will notice the step-wise process-timing of a single actor performing n sequential measurements can be represented the same as n indexed actors performing simultaneous measurements, at least with regard to internal time accounting. In order to infer the latter interpretational context, such as to preserve the common sense notion of time as distinct from social space, one would represent all n simultaneous measurements as n dimensions of X, although assumed to be generally the same in quality in such that all n actors sample the same object in the same way, yet are distinct in some orderable indexical quality. Thus, in each turn of the round time (i.e. one unit), all actors perform independent and similar measurements. It may be, as in progressive action processes, future actions are dependent on previous ones, and thus independence is only found within the sample space of a single time round. Alternatively, it may also be that the actors perform different actions, or are dependent upon each other in their interactions. Thus, the notion of actor(s) may be embedded in the space-time of the action of measurement. The embedding of a coordinated plurality of actors in the most mundane sense of ‘collective progress’ can be represented as the group action of all independent \& similar measurers completes itself in each round of time, with inter-temporalities in the process measurement process being similar but dependent on the previous one. The progressive interaction may be represented as the inducer $I^+:X(t_i) \rightarrow X(t_{i}+1)$, with the assumptions of similarity and independence as $\hat{x_i}(t) \sim \hat{x_j}(t) \ \& \ I(\hat{x_i}(t),\hat{x_j}(t))=0$. We take $\hat{X}(t)$ to be a group of measurement actors/actions $\{ \hat{x}_i(t): i \in \pi \}$ that acts on $\Omega$ together, or simultaneously, to produce a singular measurement of one round time.

\section{Derivation of the Normal Distribution}

The question with measurement is not, \say{what is the true distribution of the object in question in nature?}, but \say{what is the distribution of the parameter I am using to measure?}. The underlying metric of the quality under investigation, itself arising due to an interaction of measurement as the distance function within the investigatory space-time, is $\mu$. As the central limit states, averages of these measurements, each having an error, will converge to normality. We can describe analytically the space of our ‘atemporal’ averaged measurements in that the rate of change of the frequency $f$ of our sample measurements $x,x_0 \in X$ by the change in the space of measuring, is inversely proportional, by constant k, to the distance from the true measurement ($\mu$) and the frequency:

\[\forall \epsilon > 0, \exists \delta(\epsilon)>0 \ s.t. \ \forall x, |x_0-x|<\delta \rightarrow \bigg| k(x_0-\mu)f(x_0) - \frac{f(x_0)-f(x)}{x_0-x}\bigg|<\epsilon \]
or in the differential form
\[\frac{df}{dx}=-k(x-\mu)f(x)\]
\[f(t)=\int_{-\infty}^{+\infty}-k(x-\mu)f(x)dx\]
the solution distribution is scaled by the constant of coefficient, C
\[f(x)=Ce^{-\frac{k}{2}{(x-\mu)}^2}\]
given the normalization of the total size of the universe of events as 1
\[ \int_{-\infty}^{\infty} f dx =1\]
thus,
\[C=\sqrt{\frac{k}{2\pi}}\]
so the total distribution is,
\[f(x)=\sqrt{\frac{k}{2\pi}}e^{-\frac{k}{2}{(x-\mu)}^2}\]
\[E(X)=\int (x-\mu)f(x)dx=\mu\]
\[\sigma^2=E(X^2)=\int {(x-\mu)}^2 f(x)dx=\frac{1}{\sqrt{k}}\]
\[so, f(x)=N\bigg(\mu,\sigma=\frac{1}{\sqrt{k}}\bigg)\]\printbibliography\end{document}[/et_pb_text][/et_pb_column][/et_pb_row][/et_pb_section]