\documentclass{article}
\usepackage{amssymb, amsmath, geometry, multicol, stmaryrd, cancel, centernot}
\usepackage[table, svgnames, dvipsnames]{xcolor}
\usepackage{makecell, cellspace, caption}
\usepackage[europeanresistors]{circuitikz}
\usepackage[a]{esvect} % voir la doc pour les différents styles
\usepackage{tikz-3dplot}
\usepackage{dsfont}

\let\overrightarrow\vv


\usetikzlibrary{arrows}
\usetikzlibrary{shapes.geometric}

\newcommand{\independant}{\perp\!\!\!\perp}
\newcommand{\parspace}{\vspace{0.1cm}}
\linespread{1.25}
\geometry{tmargin=.7in, bmargin=.75in, lmargin=.5in, rmargin = .5in}

\begin{document}
\setlength{\parindent}{0mm}

\title{Signaux Aléatoires - Statistiques d'Ordre 2}
\author{Aubin SIONVILLE}
\date{Télécom St Etienne 2023-2024}

\maketitle

\center

\section*{Lois à 2 variables aléatoires}
\begin{multicols}{3}
    \subsection*{Couple de VA}
    \[\boxed{Z(\omega) = (X(\omega), Y(\omega))}\]

    \subsection*{Loi conjointe}
    \[\boxed{p_{X,Y}(x, y) = \mathbb{P}(X = x, Y = y)}\]

    \subsection*{Loi marginale}
    \[\boxed{\mathbb{P}(X=x_i) = \sum_{j=1}^{n} p_{X,Y}(x_i, y_j)}\]

    \subsection*{Loi conditionnelle}
    \[\boxed{p_{X|Y=y}(X=x) = \frac{\mathbb{P}(X=x, Y=y)}{\mathbb{P}(Y=y)}}\]

    \subsection*{Fonction de répartition conjointe}
    \[\boxed{F_{X,Y}(x, y) = \mathbb{P}(X \leq x, Y \leq y)}\]

    \subsection*{Densité de probabilité conjointe}
    \[\boxed{f_{X,Y}(x, y) = \frac{\partial^2 F_{X,Y}(x, y)}{\partial x \partial y}}\]
\end{multicols}

\begin{multicols}{2}
    \subsection*{Moments d'ordre $(p,q)$ (cas discret)}
    \[\boxed{\mathbb{E}[X^pY^q] = \sum_{i=1}^{n} \sum_{j=1}^{m} x_i^p y_j^q p_{X,Y}(x_i, y_j)}\]
    
    \subsection*{Moments d'ordre $(p,q)$ (cas continu)}
    \[\boxed{\mathbb{E}[X^pY^q] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} x^p y^q f_{X,Y}(x, y)dxdy}\]
\end{multicols}

\vspace{1cm}

\section*{Corrélation et covariance}
\begin{multicols}{2}
    \subsection*{Corrélation}
    \[\boxed{\mathbb{E}[XY] = \int_{\mathbb{R}^2} xy f_{X,Y}(x, y)dxdy}\]


    \subsection*{Coefficient de corrélation linéaire}
    \[\boxed{\rho_{X,Y} = \frac{\mathrm{Cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{\mathbb{E}[X^cY^c]}{\sqrt{\mathbb{E}\left[(X^c)^2\right]}\sqrt{\mathbb{E}\left[(Y^c)^2\right]}}}\]
\end{multicols}
\subsection*{Covariance}
$\mathrm{Cov}(X,Y) = 0 \iff$ non corrélées\\
$\mathbb{E}[XY] = 0 \iff$ orthogonales
\[\boxed{\mathrm{Cov}(X,Y) = \mathbb{E}[X^cY^c] = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]}\]

\newpage
\section*{Autocorrélation et Intercorrélation}
\begin{multicols}{2}
    \subsection*{Autocorrélation}
    \[\boxed{r_{X\!X}(t_1, t_2) = \mathbb{E}[X_{t_1}X_{t_2}^*]}\]
    \[\boxed{r_{X\!X}(t,t) = \mathbb{E}[|X_t|^2] = \mu_{2,X}(t)}\]

    \subsection*{Autocovariance}
    \[\boxed{c_{X\!X}(t_1, t_2) = \mathbb{E}[X_{t_1}^c{X_{t_2}^{c}}^*] = r_{X\!X}(t_1, t_2) - \mathbb{E}[X_{t_1}]\mathbb{E}[X_{t_2}^*]}\]
    \[\boxed{c_{X\!X}(t,t) = \sigma_X^2(t)}\]
\end{multicols}

\vspace{0.5cm}

\section*{Intercorrélation et intercovariance}
\begin{multicols}{2}
    \subsection*{Intercorrélation}
    \[\boxed{r_{X\!Y}(t_1, t_2) = \mathbb{E}[X_{t_1}Y_{t_2}^*]}\]

    \subsection*{Intercovariance}
    \[\boxed{c_{X\!Y}(t_1, t_2) = \mathbb{E}[X_{t_1}^c{Y_{t_2}^{c}}^*] = r_{X\!Y}(t_1, t_2) - \mathbb{E}[X_{t_1}]\mathbb{E}[Y_{t_2}^*]}\]
\end{multicols}


\end{document}