IndependentIdenticallyDistributed 2004-07-01 05:33:30 2007-01-06 18:48:09 Definition identically distributed \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % making logically defined graphics %\usepackage{xypic} Two random variables $X$ and $Y$ are said to be \emph{identically distributed} if they are defined on the same probability space $(\Omega,\mathcal{F},P)$, and the distribution function $F_X$ of $X$ and the distribution function $F_Y$ of $Y$ are the same: $F_X=F_Y$. When $X$ and $Y$ are identically distributed, we write $X \stackrel{d}{=} Y$. A set of random variables $X_i$, $i$ in some index set $I$, is identically distributed if $X_i \stackrel{d}{=} X_j$ for every pair $i,j\in I$. A collection of random variables $X_i$ ($i\in I$) is said to be \emph{independent identically distributed}, if the $X_i$'s are identically distributed, and \PMlinkname{mutually independent}{Independent} (every finite subfamily of $X_i$ is independent). This is often abbreviated as \emph{iid}. For example, the interarrival times $T_i$ of a Poisson process of rate $\lambda$ are independent and each have an exponential distribution with mean $1/\lambda$, so the $T_i$ are independent identically distributed random variables. Many other examples are found in statistics, where individual data points are often assumed to realizations of iid random variables.