L2-probability-functions.tex

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

{\chaptoc\noindent\begin{minipage}[inner sep=0,outer sep=0]{0.9\linewidth}\section{Probability Density Function | \(f(x)\)}\end{minipage}}

        \begin{equation}
        \begin{split}
            \mathbb{P} (a < X \leq b) = \mathbb{P} (X \leq b) - \mathbb{P} (X \leq a) = & \overbrace{F_x (b)}^{CDF} - F_x (a) \\
            = & \int_{a}^{b} \underbrace{f(x)}_{PDF} dx
        \end{split}     
        \end{equation}
            
        \Note{
        Relationship between Cumulative Distribution Function and Probability Density Function:
        \[f_x (x) = \frac{d}{dx} F_x (X) \iff F_x (t) = \int_{-\infty}^{t} f_{x} (x) dx \]
        }

        \subsection{Properties of the Probability Density Function}

            \begin{enumerate}
                \item \textbf{Non Negativity}: the density is never negative

                    \begin{equation}
                        f_x (x) \geq 0 \ \forall x \in \mathbb{R}
                    \end{equation}
                    
                \item \textbf{Normalisation}: the area below the curve of the density function is always equal to 1

                    \begin{equation}
                        \int_{-\infty}^{\infty} f_x (x) dx = 1
                    \end{equation}
                    
            \end{enumerate}

        \Example{
        \begin{equation}\label{example-pdf-1}
                f_x(x) = \begin{cases} cx^2 & \mbox{if } -1 < x \leq 2 \\ 0 & \mbox{a.e.} \end{cases}
            \end{equation}
            \[1 = \int_{-\infty}^{\infty} f_x (x) dx =  \underbrace{\int_{-\infty}^{-1} f_x (x) dx}_{=0} + \int_{-1}^{2} f_x (x) dx + \underbrace{ \int_{2}^{\infty} f_x (x) dx}_{=0} = \int_{-1}^{2} f_x (x) dx =  \]
            \[\int_{-1}^{2} cx^2 dx =\left[c \frac{x^3}{3}\right]_{-1}^{2} = \frac{c}{3} (8+1) = 3c \rightarrow 3c = 1 \rightarrow c = \frac{1}{3} \]
            Given the function \(c \cdot x^2\), the constant \(c\) that allows the function to respect the properties of a \textit{probability density function} is \(c = \frac{1}{3}\).
            The final form of function \ref{example-pdf-1} is hence:
            \[
                f_x(x) = \begin{cases} \frac{1}{3} x^2 & \mbox{if } -1 < x \leq 2 \\ 0 & \mbox{a.e.} \end{cases}
            \]
        }

        \Example{
        Imagine there is a traffic light, where the \textcolor{ForestGreen}{green light} lasts for 20' and the \textcolor{BrickRed}{red light} lasts for 40'. Define \(X\) as the waiting time. (Note: X is neither discrete nor continuous).
            \begin{equation}\label{example-pdf-1}
                F_x(t) = \begin{cases} 0 & \mbox{if } t<0 \\ \clubsuit & \mbox{if } 0 \leq t < 40 \\ 1 & \mbox{if } t \geq 40 \end{cases}
            \end{equation}
            \[
            \clubsuit = \mathbb{P} (x \leq t) = \mathbb{P} (x \leq t | \textcolor{BrickRed}{R}) \cdot \mathbb{P} (\textcolor{BrickRed}{R}) + \mathbb{P} (x \leq t | \textcolor{ForestGreen}{G}) \cdot \mathbb{P} (\textcolor{ForestGreen}{G}) = 
            \]
            \[
            \frac{t}{40} \cdot \overbrace{\frac{40}{20+40}}^{prob. \ light=\textcolor{BrickRed}{R}} + 1 \cdot \overbrace{\frac{20}{20+40}}^{prob. \ light = \textcolor{ForestGreen}{G}} =  
            \]
            \[
            \clubsuit = \frac{1}{3} \cdot \frac{t+20}{60}
            \]
        }

\section{Expected Value}

    The expected value of a probability function can be described as:

        \begin{equation}
            E(x)= \int_{0}^{+\infty} (1-F_x(t)) dt - \int_{-\infty}^{0} F_{x}(t) dt \begin{cases}
                \sum x_i p_x (I) & \mbox{if $X$ is discrete} \\ \int_{-\infty}^{+\infty} x f_x(x) dx & \mbox{if $X$ is continuous} \end{cases}
        \end{equation}

    \Example{
        When X = \textit{number of tiles}:
        \[E(x) = 1 \cdot \frac{1}{4} + 2 \cdot \frac{1}{2} + 3 \cdot \frac{4-\pi}{16} + 4 \cdot \frac{\pi}{16} = 2+ \frac{\pi}{16} \]
    }

    \Example{
        When X = \textit{distance from centre}:
        \[E(x) = \int_{-\infty}^{+\infty} x f_x(x) dx  \]
    }

    \Example{
        When X = \textit{waiting time at the traffic light}:
        \[E(x) = \int_{-\infty}^{+\infty} x f_x(x) dx  \]
    }

        
        fine pagine