L1-probability.tex

\setchapterabstract{}
\chapter{Probability}
\vspace{-1.5cm}

{\chaptoc\noindent\begin{minipage}[inner sep=0,outer sep=0]{0.9\linewidth}\section{Elementary Probability}\end{minipage}}

        \subsubsection{Permutation}

        \Definition{
        any alignment in $n$ places
                    \begin{equation}
                        P_n = n!
                    \end{equation}
        }{Permutation}

        \subsubsection{Disposition}

        \Definition{
        Any alignment in $k$ places
        \begin{enumerate}
                        \item \textbf{Simple}:
                            \begin{equation}
                                D_{n,k} = n \cdot (n-1) \cdot \cdots \cdot (n-k-1) = \frac{n!}{(n-k)!}
                            \end{equation}
                        \item \textbf{With repetition}:
                            \begin{equation}
                                D_{n,k}^* = n \cdot n \cdot \cdots \cdot n = n^k 
                            \end{equation}
                    \end{enumerate}
        }{Disposition}

        \subsubsection{Combination}

        \Definition{
        any alignment of $k$ objects from $n$
                    \begin{equation}
                        C_{n,k} = \frac{n!}{k!(n-k)!} = \binom{n}{k}
                    \end{equation}
        }{Combination}

\section{Conditional Probability}

    In the probability space \( ( \Omega, A, \mathbb{P}) \), conditional probability can be written as

    \begin{equation}
        \mathbb{P} (E|F) = \frac{\mathbb{P} (E \cap F)}{\mathbb{P} (F)}
    \end{equation}

    \noindent
    with \(\mathbb{P} (F) > 0\)

        \subsubsection{Conditional Probability as Intersection of Events}

            \begin{equation}
            \begin{split}
                \mathbb{P} (E \cap F) = & \mathbb{P} (F) \cdot \mathbb{P} (E | F) \\
                \mathbb{P} (E_1 \cap E_2 \cap \cdots \cap E_n ) = & \mathbb{P} (E_1) \cdot \mathbb{P} (E_2 | E_1) \cdot \mathbb{P} (E_3 | E_2 \cap E_1) \cdots \mathbb{P} (E_n | E_1 \cap E_2 \cap \cdots \cap E_{n-1})
            \end{split}
            \end{equation}

        \subsubsection{Law of Total Probability}

            Given \(E_1, \ E_2, \ \cdots \ E_n\) partitions of \(\Omega\) and the conditions

            \begin{enumerate}
                \item \(E_i \cap E_j = \emptyset \forall i \neq j\)
                \item \(E_1 \cup E_2 \cup \cdots \cup E_n = \Omega\)
            \end{enumerate}

            \noindent
            We can state that:

            \begin{equation}
                \mathbb{P} (F) = \sum^{n}_{i=1} \mathbb{P} (F|Ei) \mathbb{P} (E_i)
            \end{equation}

        \subsubsection{Bayes Formula}

            Given 

            \begin{equation}
                \mathbb{P} (E_n | F) = \frac{\mathbb{P} (F| E_n) \mathbb{P} (E_n)}{\sum^{n}_{i=1} \mathbb{P} (F|Ei) \mathbb{P} (E_i)} \rightarrow \mathbb{P} (A|P) \frac{\mathbb{P} (B|A) \mathbb{P} (A)}{\mathbb{P} (B)}
            \end{equation}

\section{Independent Events}

        \subsubsection{Stochastic Independence}

            \Definition{
            Given the probability space \( ( \Omega, A, \mathbb{P} ) \), two events \(E, F \in A\) are said to be \textbf{stocastically independent} if \(\mathbb{P} (E) \cap \mathbb{P} (F) = \mathbb{P} (E) \cdot \mathbb{P} (F)\)
            \begin{equation}
                \mathbb{P} (E|F) = \mathbb{P} (E) \iff \mathbb{P} (F|E) = \mathbb{P} (F)
            \end{equation}
            }{Stochastic Independence}
            

        \subsubsection{Conditional Stochastic Independence}

            \Definition{
            Given the probability space \( ( \Omega, A, \mathbb{P} ) \) and the events \(A, B, F \), \textbf{Conditional Stochastic Independence} can be characterised as:
            \begin{equation}
                \mathbb{P} (A \cap B | F) = \mathbb{P} (A|F) \cdot \mathbb{P} (B|F)
            \end{equation}
            }{Conditional Stochastic Independence}

            \Warning{
            Stochastic Independence \textbf{does not imply} Conditional Stochastic Independence, and vice versa.
            }
            

\section{Random Variables}

        Given the sample space of tossing a dice \(\Omega = \{1,2,3,4,5,6\} \), we can define:

            \[X: \Omega \rightarrow \mathbb{R} \]

        \Remark{
        In the equation above:
                \begin{itemize}
                    \item if \(\Omega\) is finite \(\rightarrow\) it is a random variable
                    \item if \(\Omega\) is discrete \(\rightarrow\) it is \textbf{not} a random variable 
                \end{itemize}
        }

        \noindent
        Properties to be considered a random variable:

            \begin{itemize}
                \item \(\forall t \in \mathbb{R}\)

                    \( \{ \omega \in \Omega : X (\omega) \leq t \} = E_t \in A \)

                    With \(X\) being \textbf{Borel measurable}

                \item 

                    \( \mathbb{P} (E_t) = \mathbb{P} (\omega \in \Omega : X (\omega) \leq t ) = \mathbb{P} (X (\omega) \leq t) = \mathbb{P} (X \leq t)  0 F_x (t)  \)
 
                \item \(F_x : \mathbb{R} \rightarrow \mathbb{R}\), with \(F_x(t)\) being the distribution function of the variable.

                    \( \lim _{t \to - \infty} F(t) = 0 \)    

                    \( \lim _{t\to\infty} F(t) = 1\)

                    \begin{itemize}
                        \item If \(X\) is discrete, the graph will possess the following properties:
                            \begin{enumerate}
                                \item Stepwise
                                \item Non-decreasing 
                                \item \(F_x\) continuous from the right
                            \end{enumerate}
                        \item If \(X\) is \textcolor{dblue}{absolutely} continuous
                            \begin{enumerate}
                                \item \(F\) is continuous
                            \end{enumerate}
                    \end{itemize}
            \end{itemize}

            \begin{figure}[h]
                \centering
                \includegraphics[width=0.90\linewidth]{X_visual_discrete.png}
                \caption{Visual representation of an example where \(X\) is a \textbf{discrete} random variable}
                \label{fig:X_visual_discrete}
            \end{figure}

            \begin{figure}[h]
                \centering
                \includegraphics[width=0.90\linewidth]{X_visual_absolutely_continuous.png}
                \caption{Visual representation of an example where \(X\) is an \textbf{absolutely continuous} random variable}
                \label{fig:X_visual_absolutely_continuous}
            \end{figure}