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\Large{\textbf{Implementations of Probability Theory}}\\

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\Large{Independent Study Report}\\
\large{Andrew Simonson}

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\large{Compiled on: \today}\\
 
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\section{Objective}
yada yada yah I started this independent study for my own selfish gain

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\section{Units}
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\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<X\leq 1\\
        0&otherwise
    \end{cases}
\end{equation*}

\subsubsection{Limit Theorems}
\subsubsection*{Law of Large Numbers}
The Law of Large Numbers states that as the number of independent random samples increases, the average of the samples' 
means will approach the true mean of the population. 
\[\text{true average}\approx \frac{1}{n} \sum_{i=1}^{n} X_{i} \qquad\text{as }n \rightarrow \infty\]
\subsubsection*{Central Limit Theorem}
The Central Limit Theorem states that the sampling distribution of a sample mean is a normal distribution even when the 
population distribution is not normal.
\[
\frac{\sqrt{n} \left( \bar{X}_n - \mu \right)}{\sigma} \xrightarrow{d} N(0, 1),
\]
\[
\text{Where \( \bar{X}_n = \frac{1}{n} \sum_{i=1}^{n}\), \( X_i \) is the sample mean, and \( N(0, 1) \) is a standard normal distribution.}
\]
This is a challenging to understand solely as an equation.  As an example, take a sample of two six-sided dice rolls and average their numbers.  
The more sample averages taken, the more they will resemble a normal distribution where the majority of samples average around 3.

\subsubsection{Confidence}
Confidence is described using a confidence interval, which is a range of values that the true value is expected to be in, and its associated confidence level, 
which is a probability (expressed as a percentage) that the true value is in the confidence interval.

% Confidence intervals can be calculated with z-tests, t-tests.  Go into parametric vs non-parametric

\subsubsection{Statistical Inference}
Statistical Inference is any data analysis to draw conclusions from a sample to make assertions about the population.  
Methods include estimation via averages and confidence intervals, and hypothesis testing, which attempts to invalidate (never \textit{validate}) a hypothesis.

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