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\title{Signaux Aléatoires - Statistiques d'Ordre 1}
\author{Aubin SIONVILLE}
\date{Télécom St Etienne 2023-2024}

\maketitle

\center

\section*{Moments d'une variable aléatoire}
\begin{multicols}{2}
    \subsection*{Moment d'ordre $m$}
    \[\boxed{\mu_m = \mathbb{E}[X^m]}\]

    \subsection*{Moment centré d'ordre $m$}
    \[\boxed{\mu_m^c = \mathbb{E}[(X^c)^m] = \mathbb{E}[(X - \mu)^m]}\]
\end{multicols}

\subsection*{Variance}
\[\boxed{\sigma^2 = \mathbb{E}[(X - \mu)^2] = \mu_2^c}\]

\vspace{1cm}

\section*{Extension aux signaux aléatoires}
\begin{multicols}{2}
    \subsection*{Moment d'ordre $m$}
    \[\boxed{\mu_{m,X}[n] = \mathbb{E}[X_n^m]}\]

    \subsection*{Moment centré d'ordre $m$}
    \[\boxed{\mu_{m,X}^c[n] = \mathbb{E}[(X_n^c)^m] = \mathbb{E}[(X_n - \mu_X[n])^m]}\]
\end{multicols}

\subsection*{Statistique d'ordre 1}
\[\boxed{\mu_{1,X}^c[n] = \mu_X[n] = \mathbb{E}[X_n]}\]


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